Shixin Zhu

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New quantum MDS codes derived from constacyclic codes

2015-01-06

Liqi Wang , Shixin Zhu

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Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

2019-10-30

Weijun Fang , Fang‐Wei Fu , Lanqiang Li +1 more

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their … In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length n > q + 1 are also obtained.

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Entanglement-assisted quantum MDS codes constructed from constacyclic codes

2018-09-06

Xiaojing Chen , Shixin Zhu , Xiaoshan Kai

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Repeated-root constacyclic codes of length 3lp and their dual codes

2016-09-13

Li Liu , Lanqiang Li , Xiaoshan Kai +1 more

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A Class of Narrow-Sense BCH Codes

2019-05-24

Shixin Zhu , Zhonghua Sun , Xiaoshan Kai

BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their … BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers. 1) The dimension of narrow-sense BCH codes of length ((q m -1)/λ) with designed distance 2 ≤ δ ≤ ((q Γ(m+1)/2⌉ - 1)/(λ) + 1) is settled, where λ is any factor of (q - 1). 2) The weight distributions of two classes of narrow-sense BCH codes of length ((q m - 1)/2) with designed distance δ = (((q - 1)q m-1 - q ⌊(m-1)12⌋ - 1)/2) and δ = (((q - 1)q m-1 - q ⌊(m+1)/2⌋ - 1)/2) are determined. 3) The weight distribution of a class of BCH codes of length ((q m - 1)/(q - 1)) is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized.

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On LCD Negacyclic Codes over Finite Fields

2017-11-29

Binbin Pang , Shixin Zhu , Zhonghua Sun

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New quantum codes from dual-containing cyclic codes over finite rings

2016-08-22

Yongsheng Tang , Shixin Zhu , Xiaoshan Kai +1 more

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Cyclic codes and some new entanglement-assisted quantum MDS codes

2021-09-06

Xiaojing Chen , Shixin Zhu , Wan Jiang

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A Class of Optimal Cyclic Codes With Two Zeros

2019-06-06

Dengchuan Liao , Xiaoshan Kai , Shixin Zhu +1 more

Let m > 2 be an integer and p be an odd prime. We explore the minimum distance of p-ary cyclic codes of length n = 2(p m … Let m > 2 be an integer and p be an odd prime. We explore the minimum distance of p-ary cyclic codes of length n = 2(p m - 1)/(p - 1) with two zeros. A sufficient condition for such cyclic codes with minimum distance at least three is obtained. A class of optimal p-ary cyclic codes with minimum distance four are presented. Four explicit constructions for such optimal cyclic codes are provided. The weight distribution of the dual of the cyclic code in the first construction is given.

Cyclic codes over R=Fp+uFp+⋯+uk−1Fp with length psn

2010-10-30

Mu Han , Youpei Ye , Shixin Zhu +2 more

On the construction of optimal asymmetric quantum codes

2014-04-01

Liqi Wang , Shixin Zhu

Constacyclic codes are important classes of linear codes that have been applied to the construction of quantum codes. Six new families of asymmetric quantum codes derived from constacyclic codes are … Constacyclic codes are important classes of linear codes that have been applied to the construction of quantum codes. Six new families of asymmetric quantum codes derived from constacyclic codes are constructed in this paper. Moreover, the constructed asymmetric quantum codes are optimal and different from the codes available in the literature.

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The Hull of Two Classical Propagation Rules and Their Applications

2023-06-20

Yang Li , Shixin Zhu , Edgar Martı́nez-Moro

In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(u,u+v)$ </tex-math></inline-formula> -construction and the … In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(u,u+v)$ </tex-math></inline-formula> -construction and the direct sum construction. Some new criteria for the resulting codes derived from these two propagation rules being self-dual, self-orthogonal, or linear complementary dual (LCD) codes are given. As applications, we employ the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(u,u+v)$ </tex-math></inline-formula> -construction to obtain (almost) self-orthogonal codes; employ the direct sum construction to provide lower bounds on the minimum distance of FSD (LCD) codes; and employ both these two constructions to derive linear codes with prescribed hull dimensions. Many (almost) optimal codes are presented. In particular, a family of binary almost Euclidean self-orthogonal Griesmer codes is constructed. We also obtain many binary, ternary Euclidean and quaternary Hermitian FSD LCD codes of larger lengths and improve some lower bounds on the minimum distance of known ternary Euclidean LCD codes.

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New entanglement-assisted quantum MDS codes with length $$n=\frac{q^2+1}{5}$$

2020-06-10

Shixin Zhu , Wan Jiang , Xiaojing Chen

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On MDS codes with galois hulls of arbitrary dimensions

2022-12-17

Yang Li , Shixin Zhu , Ping Li

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Some new bounds on LCD codes over finite fields

2020-01-09

Binbin Pang , Shixin Zhu , Xiaoshan Kai

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On Self‐dual and LCD Double Circulant Codes over a Non‐chain Ring*

2019-09-01

Ting Yao , Shixin Zhu , Xiaoshan Kai

In this paper, we study double circulant codes of length 2n over the non-chain ring R = 𝔽q + v𝔽q + v2𝔽q; where q is an odd prime power and … In this paper, we study double circulant codes of length 2n over the non-chain ring R = 𝔽q + v𝔽q + v2𝔽q; where q is an odd prime power and v3 = v. Exact enumerations of self-dual and LCD double circulant codes of length 2n over R are derived. When n is an odd prime, using random coding, we obtain families of asymptotically good self-dual and LCD codes of length 6n over 𝔽q.

MDS codes with Euclidean and Hermitian hulls of flexible dimensions and their applications to EAQECCs

2023-03-22

Yang Li , Ruhao Wan , Shixin Zhu

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Planarity of mappings<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Tr</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mi>α</mml:mi><mml:mn>2</mml:mn></mml:mfrac><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>on finite fields

2013-04-18

Minghui Yang , Shixin Zhu , Keqin Feng

Cyclic DNA codes over F2+uF2+vF2+uvF2

2015-01-01

Shixin Zhu , Xiaojing Chen

In this work, we study the structure of cyclic DNA codes of arbitrary lengths over the ring R=F2+uF2+vF2+uvF2 and establish relations to codes over R1=F2+uF2 by defining a Gray map … In this work, we study the structure of cyclic DNA codes of arbitrary lengths over the ring R=F2+uF2+vF2+uvF2 and establish relations to codes over R1=F2+uF2 by defining a Gray map between R and R1^2 where R1 is the ring with 4 elements. Cyclic codes of arbitrary lengths over R satisfied the reverse constraint and the reverse-complement constraint are studied in this paper. The GC content constraint is considered in the last.

Several classes of Galois self-orthogonal MDS codes and related applications

2023-07-27

Yang Li , Yunfei Su , Shixin Zhu +2 more

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Several classes of optimal ternary cyclic codes

2017-01-01

Lanqiang Li , Li Liu , Shixin Zhu

Cyclic codes have efficient encoding and decoding algorithms over finite fields, so that they have practical applications in communication systems, consumer electronics and data storage systems. The objective of this … Cyclic codes have efficient encoding and decoding algorithms over finite fields, so that they have practical applications in communication systems, consumer electronics and data storage systems. The objective of this paper is to give eight new classes of optimal ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$, according to a result on the non-existence of solutions to a certain equation over $F_{3^m}$. It is worth noticing that some recent conclusions on such optimal ternary cyclic codes are some special cases of our work. More importantly, three of the nine open problems proposed by Ding and Helleseth in [8] are solved completely. In addition, another one among the nine open problems is also promoted.

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The ranks of cyclic and negacyclic codes over the finite ring R

2008-01-01

Shixin Zhu , Minjia Shi

On the constructions of $n$-cycle permutations

2020-01-01

Yuting Chen , Liqi Wang , Shixin Zhu

Any permutation polynomial is an $ n $-cycle permutation. When $n$ is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. … Any permutation polynomial is an $ n $-cycle permutation. When $n$ is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. These permutations have important applications in cryptography and coding theory. Inspired by the AGW Criterion, we propose criteria for $ n $-cycle permutations, which mainly are of the form $ x^rh(x^s) $. We then propose unified constructing methods including recursive ways and a cyclotomic way for $ n $-cycle permutations of such form. We demonstrate our approaches by constructing three classes of explicit triple-cycle permutations with high index and two classes of $ n $-cycle permutations with low index.

BCH codes with larger dimensional hull

2023-08-17

Binbin Pang , Shixin Zhu , Tian Yang +1 more

A generalization of the Tang-Ding binary cyclic codes

2025-01-21

Ling Li , Minjia Shi , Sihui Tao +4 more

A Class of Narrow-Sense BCH Codes

2019-01-01

Shixin Zhu , Zhonghua Sun , Xiaoshan Kai

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Repeated-root constacyclic codes of length $3lp^{s}$ and their dual codes

2016-01-01

Li Liu , Lanqiang Li , Xiaoshan Kai +1 more

Let $p\neq3$ be any prime and $l\neq3$ be any odd prime with $gcd(p,l)=1$. $F_{q}^{*}=\langle\xi\rangle$ is decomposed into mutually disjoint union of $gcd(q-1,3lp^{s})$ coset over the subgroup $\langle\xi^{3lp^{s}}\rangle$, where $\xi$ is … Let $p\neq3$ be any prime and $l\neq3$ be any odd prime with $gcd(p,l)=1$. $F_{q}^{*}=\langle\xi\rangle$ is decomposed into mutually disjoint union of $gcd(q-1,3lp^{s})$ coset over the subgroup $\langle\xi^{3lp^{s}}\rangle$, where $\xi$ is a primitive $(q-1)$th root of unity. We classify all repeated-root constacyclic codes of length $3lp^{s}$ over the finite field $F_{q}$ into some equivalence classes by the decomposition, where $q=p^{m}$, $s$ and $m$ are positive integers. According to the equivalence classes, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length $3lp^{s}$ over $F_{q}$ and their dual codes. Self-dual cyclic(negacyclic) codes of length $3lp^{s}$ over $F_{q}$ exist only when $p=2$. And we give all self-dual cyclic(negacyclic) codes of length $3l2^{s}$over $F_{2^{m}}$ and its enumeration.

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Repeated-root constacyclic codes of length 6lmpn

2021-09-27

Tingting Wu , Shixin Zhu , Li Liu +1 more

Let <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> be a finite field with character <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, the multiplicative group <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_{q}^{*} = \mathbb{F}_{q}\setminus\{0\} $\end{document}</tex-math></inline-formula> is decomposed … Let <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> be a finite field with character <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, the multiplicative group <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_{q}^{*} = \mathbb{F}_{q}\setminus\{0\} $\end{document}</tex-math></inline-formula> is decomposed into a mutually disjoint union of <inline-formula><tex-math id="M4">\begin{document}$ \gcd(6l^mp^n,q-1) $\end{document}</tex-math></inline-formula> cosets over subgroup <inline-formula><tex-math id="M5">\begin{document}$ <\xi^{6l^mp^n}> $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is a primitive element of <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>. Based on the decomposition, the structure of constacyclic codes of length <inline-formula><tex-math id="M8">\begin{document}$ 6l^mp^n $\end{document}</tex-math></inline-formula> over finite field <inline-formula><tex-math id="M9">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> and their duals is established in terms of their generator polynomials, where <inline-formula><tex-math id="M10">\begin{document}$ p\neq{3} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ l\neq{3} $\end{document}</tex-math></inline-formula> are distinct odd primes, <inline-formula><tex-math id="M12">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula> are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length <inline-formula><tex-math id="M14">\begin{document}$ 6l^mp^n $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M15">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>.

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$\mathbb{Z}_p\mathbb{Z}_p[u]$-additive codes

2015-01-01

Zhenliang Lu , Shixin Zhu

In this paper, we study $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive codes, where $p$ is prime and $u^{2}=0$. In particular, we determine a Gray map from $ \mathbb{Z}_p\mathbb{Z}_p[u]$ to $\mathbb{Z}_p^{ α+2 β}$ and study generator … In this paper, we study $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive codes, where $p$ is prime and $u^{2}=0$. In particular, we determine a Gray map from $ \mathbb{Z}_p\mathbb{Z}_p[u]$ to $\mathbb{Z}_p^{ α+2 β}$ and study generator and parity check matrices for these codes. We prove that a Gray map $Φ$ is a distance preserving map from ($\mathbb{Z}_p\mathbb{Z}_p[u]$,Gray distance) to ($\mathbb{Z}_p^{α+2β}$,Hamming distance), it is a weight preserving map as well. Furthermore we study the structure of $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive cyclic codes.

Gray map over ring F_2+uF_2

2010-01-01

Shixin Zhu

In this paper,the Gray map φ form R=F2 +uF2 to F2 is given,and the Gray map is proved to be weight-preserving and distance-preserving.The binary image of a linear code C … In this paper,the Gray map φ form R=F2 +uF2 to F2 is given,and the Gray map is proved to be weight-preserving and distance-preserving.The binary image of a linear code C over ring R,called φ(C) is Hamming distance invariant,if C is Lee distance invariant.

A note on complete classification of $(δ+αu^2)$-constacyclic codes of length $p^k$ over $\F_{p^m}+u\F_{p^m}+u^2\F_{p^m}$

2017-01-01

R. Sobhani , Zhonghua Sun , Wang Li-qi +1 more

For units $δ$ and $α$ in $\F_{p^m}$, the structure of $(δ+αu^2)$-constacyclic codes of length $p^k$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$ is studied and self-dual $(δ+αu^2)$-constacyclic codes are analyzed. For units $δ$ and $α$ in $\F_{p^m}$, the structure of $(δ+αu^2)$-constacyclic codes of length $p^k$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$ is studied and self-dual $(δ+αu^2)$-constacyclic codes are analyzed.

Some results of linear codes over the ring ℤ 4 +uℤ 4 +vℤ 4 +uvℤ 4 .

2016-01-01

Ping Li , Xuemei Guo , Shixin Zhu

Entanglement-assisted quantum MDS codes constructed from constacyclic codes

2018-01-01

Xiaojing Chen , Shixin Zhu , Xiaoshan Kai

Recently, entanglement-assisted quantum error correcting codes (EAQECCs) have been constructed by cyclic codes and negacyclic codes. In this paper, by analyzing the cyclotomic cosets in the defining set of constacyclic … Recently, entanglement-assisted quantum error correcting codes (EAQECCs) have been constructed by cyclic codes and negacyclic codes. In this paper, by analyzing the cyclotomic cosets in the defining set of constacyclic codes, we constructed three classes of new EAQECCs which satisfy the entanglement-assisted quantum Singleton bound. Besides, three classes of EAQECCs with maximal entanglement from constacyclic codes are constructed in the meanwhile.

A family of constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ and application to quantum codes

2017-12-06

Yongsheng Tang , Ting Yao , Shixin Zhu +1 more

We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over … We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over $\mathbb{F}_{2}.$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ of length $n$ is permutation equivalent to a binary quasi-cyclic code of index $m.$ Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.$

On cyclic codes of length $2^e$ over finite fields

2018-01-01

Binbin Pang , Shixin Zhu , Ping Li

Professor Cunsheng Ding gave cyclotomic constructions of cyclic codes with length being the product of two primes. In this paper, we study the cyclic codes of length $n=2^e$ and dimension … Professor Cunsheng Ding gave cyclotomic constructions of cyclic codes with length being the product of two primes. In this paper, we study the cyclic codes of length $n=2^e$ and dimension $k=2^{e-1}$. Clearly, Ding's construction is not hold in this place. We describe two new types of generalized cyclotomy of order two, which are different from Ding's. Furthermore, we study two classes of cyclic codes of length $n$ and dimension $k$. We get the enumeration of these cyclic codes. What's more, all of the codes from our construction are among the best cyclic codes. Furthermore, we study the hull of cyclic codes of length $n$ over $\mathbb{F}_q$. We obtain the range of $\ell=\dim({\rm Hull}(C))$. We construct and enumerate cyclic codes of length $n$ having hull of given dimension.

MDS codes with Hermitian hulls of arbitrary dimensions and their quantum error correction.

2018-12-21

Lanqiang Li , Shixin Zhu , Li Liu +1 more

The Hermitian hull of linear codes plays an important role in coding theory and quantum coding theory. In this paper, we first construct some infinite classes of MDS codes over … The Hermitian hull of linear codes plays an important role in coding theory and quantum coding theory. In this paper, we first construct some infinite classes of MDS codes over finite field by considering generalized Reed-Solomon codes or extended generalized Reed- Solomon codes and determine their Hermitian hulls. The results indicate that these codes constructed have Hermitian hulls of (almost) arbitrary dimensions. Furthermore, based on these MDS codes constructed with Hermitian hulls of arbitrary dimensions, we obtain several infinite classes of entanglement-assisted quantum error-correcting (EAQEC) MDS codes whose maximally entangled states c can take almost all values. Moreover, most of the EAQEC MDS codes constructed are new in the sense that their parameters are different from all the previously known ones.

New quantum codes from dual-containing cyclic codes over finite rings

2016-01-01

Yongsheng Tang , Shixin Zhu , Xiaoshan Kai +1 more

Let $R=\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}+\cdots+u^{k}\mathbb{F}_{2^{m}}$ , where $\mathbb{F}_{2^{m}}$ is a finite field with $2^{m}$ elements, $m$ is a positive integer, $u$ is an indeterminate with $u^{k+1}=0.$ In this paper, we propose the constructions … Let $R=\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}+\cdots+u^{k}\mathbb{F}_{2^{m}}$ , where $\mathbb{F}_{2^{m}}$ is a finite field with $2^{m}$ elements, $m$ is a positive integer, $u$ is an indeterminate with $u^{k+1}=0.$ In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over $R$. A new Gray map over $R$ is defined and a sufficient and necessary condition for the existence of dual-containing cyclic codes over $R$ is given. A new family of $2^{m}$-ary quantum codes is obtained via the Gray map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$ Furthermore, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$

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New quantum MDS codes derived from constacyclic codes

2014-01-01

Liqi Wang , Shixin Zhu

Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. It is very hard to construct quantum MDS codes with relatively large minimum distance. In this paper, based on … Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. It is very hard to construct quantum MDS codes with relatively large minimum distance. In this paper, based on classical constacyclic codes, we construct two classes of quantum MDS codes with parameters $$[[\lambda(q-1),\lambda(q-1)-2d+2,d]]_q$$ where $2\leq d\leq (q+1)/2+\lambda-1$, and $q+1=\lambda r$ with $r$ even, and $$[[\lambda(q-1),\lambda(q-1)-2d+2,d]]_q$$ where $2\leq d\leq (q+1)/2+\lambda/2-1$, and $q+1=\lambda r$ with $r$ odd. The quantum MDS codes exhibited here have parameters better than the ones available in the literature.

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On the Construction of Optimal Asymmetric Quantum Codes

2014-01-01

Liqi Wang , Shixin Zhu

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New entanglement-assisted quantum MDS codes with length $n=\frac{q^2+1}5$

2019-01-01

Shixin Zhu , Wan Jiang , Xiaojing Chen

The entanglement-assisted stabilizer formalism can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs). In this work, we construct some new entanglement-assisted quantum MDS (EAQMDS) codes with … The entanglement-assisted stabilizer formalism can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs). In this work, we construct some new entanglement-assisted quantum MDS (EAQMDS) codes with length $n=\frac{q^2+1}5$ from cyclic codes. Compared with all the previously known parameters with the same length, all of them have flexible parameters and larger minimum distance.

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Cyclic codes and some new entanglement-assisted quantum MDS codes

2020-01-01

Wan Jiang , Shixin Zhu , Xiaojing Chen

Entanglement-assisted quantum error correcting codes (EAQECCs) play a significant role in protecting quantum information from decoherence and quantum noise. Recently, constructing entanglement-assisted quantum maximum distance separable (EAQMDS) codes with flexible … Entanglement-assisted quantum error correcting codes (EAQECCs) play a significant role in protecting quantum information from decoherence and quantum noise. Recently, constructing entanglement-assisted quantum maximum distance separable (EAQMDS) codes with flexible parameters has received much attention. In this work, four families of EAQMDS codes with a more general length are presented. And the method of selecting defining set is different from others. Compared with all the previously known results, the EAQMDS codes we constructed have larger minimum distance. All of these EAQMDS codes are new in the sense that their parameters are not covered by the quantum codes available in the literature.

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Construction of self-dual MDR cyclic codes over finite chain rings

2022-06-12

Jian Yuan , Shixin Zhu , Xiaoshan Kai

New MDS self-dual codes over finite fields $\F_{r^2}$

2022-01-01

Ruhao Wan , Shixin Zhu

MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue … MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue in coding theory. In this paper, we develop the existing theory and construct six new classes of MDS self-dual codes. Together with our constructions, the proportion of all known MDS self-dual codes relative to possible MDS self-dual codes generally exceed 57\%. As far as we know, this is the largest known ratio. Moreover, some new families of MDS self-orthogonal codes and MDS almost self-dual codes are also constructed.

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Construction of MDS self-dual codes from generalized Reed-Solomon codes

2022-01-01

Ruhao Wan , Shixin Zhu , Jin Li

MDS codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over … MDS codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field $F_q$ is completely solved for $q$ is even. In this paper, for finite field with odd characteristic, we construct some new classes of MDS self-dual codes by (extended) generalized Reed-Solomon codes.

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On Galois hulls of linear codes and new entanglement-assisted quantum error-correcting codes

2022-01-01

Yang Li , Shixin Zhu

The Galois hull of a linear code is the intersection of itself and its Galois dual code, which has aroused the interest of researchers in these years. In this paper, … The Galois hull of a linear code is the intersection of itself and its Galois dual code, which has aroused the interest of researchers in these years. In this paper, we study Galois hulls of linear codes. Firstly, the symmetry of the dimensions of Galois hulls of linear codes is found. Some new necessary and sufficient conditions for linear codes being Galois self-orthogonal codes, Galois self-dual codes, and Galois linear complementary dual codes are characterized. Then, we propose explicit methods to construct Galois self-orthogonal codes of larger length from given Galois self-orthogonal codes. As an application, linear codes of larger length with Galois hulls of arbitrary dimensions are further derived. Focusing on the Hermitian inner product, two new classes of Hermitian self-orthogonal maximum distance separable (MDS) codes are also constructed. Finally, applying all the results to the construction of entanglement-assisted quantum error-correcting codes (EAQECCs), many new $q$-ary or $\sqrt{q}$-ary EAQECCs and MDS EAQECCs with rates greater than or equal to $\frac{1}{2}$ and positive net rates can be obtained. Moreover, the minimum distance of many $\sqrt{q}$-ary MDS EAQECCs of length $n>\sqrt{q}+1$ is greater than or equal to $\lceil \frac{\sqrt{q}}{2} \rceil$.

Several classes of Galois self-orthogonal MDS codes and related applications

2022-01-01

Yang Li , Yunfei Su , Shixin Zhu

Let $q=p^h$ be a prime power and $e$ be an integer with $0\leq e\leq h-1$. $e$-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes ($e=0$) and Hermitian self-orthogonal codes ($e=\frac{h}{2}$ … Let $q=p^h$ be a prime power and $e$ be an integer with $0\leq e\leq h-1$. $e$-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes ($e=0$) and Hermitian self-orthogonal codes ($e=\frac{h}{2}$ and $h$ is even). In this paper, we propose two general methods to construct $e$-Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of $e$-Galois self-orthogonal (extended) GRS codes with odd $q$ and $2e\mid h$ are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applications, new $e'$-Galois self-orthogonal maximum distance separable (MDS) codes for all possible $e'$ satisfying $0\leq e'\leq h-1$, new $e$-Galois self-orthogonal MDS codes via the shortened codes, and new MDS codes with prescribed dimensional $e$-Galois hull via the punctured codes are derived. Moreover, some new $\sqrt{q}$-ary quantum MDS codes with lengths greater than $\sqrt{q}+1$ and minimum distances greater than $\frac{\sqrt{q}}{2}+1$ are obtained.

On MDS Codes With Galois Hulls of Arbitrary Dimensions

2022-09-13

Yang Li , Shixin Zhu , Ping Li

Abstract The Galois hulls of linear codes are a generalization of the Euclidean and Hermitian hulls of linear codes. In this paper, we study the Galois hulls of (extended) GRS … Abstract The Galois hulls of linear codes are a generalization of the Euclidean and Hermitian hulls of linear codes. In this paper, we study the Galois hulls of (extended) GRS codes and present several new constructions of MDS codes with Galois hulls of arbitrary dimensions via (extended) GRS codes. Two general methods of constructing MDS codes with Galois hulls of arbitrary dimensions by Hermitian or general Galois self-orthogonal (extended) GRS codes are given. Using these methods, some MDS codes with larger dimensions and Galois hulls of arbitrary dimensions can be obtained and relatively strict conditions can also lead to many new classes of MDS codes with Galois hulls of arbitrary dimensions. 2010 MSC: 12E20, 81p70

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MDS Codes with Euclidean and Hermitian Hulls of Flexible Dimensions and Their Applications to EAQECCs

2022-01-01

Yang Li , Ruhao Wan , Shixin Zhu

The hull of a linear code is the intersection of itself with its dual code with respect to certain inner product. Both Euclidean and Hermitian hulls are of theorical and … The hull of a linear code is the intersection of itself with its dual code with respect to certain inner product. Both Euclidean and Hermitian hulls are of theorical and practical significance. In this paper, we construct several new classes of MDS codes via (extended) generalized Reed-Solomon (GRS) codes and determine their Euclidean or Hermitian hulls. Specifically, four new classes of MDS codes with Hermitian hulls of flexible dimensions and six new classes of MDS codes with Euclidean hulls of flexible dimensions are constructed. For the former, we further construct four new classes of entanglement-assisted quantum error-correcting codes (EAQECCs) and four new classes of MDS EAQECCs of length $n>q+1$. For the latter, we also give some examples on Euclidean self-orthogonal and one-dimensional Euclidean hull MDS codes.

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Research on Hermitian self-dual codes, GRS codes and EGRS codes

2022-01-01

Ruhao Wan , Shixin Zhu

MDS self-dual codes have nice algebraic structures, theoretical significance and practical implications. In this paper, we present three classes of $q^2$-ary Hermitian self-dual (extended) generalized Reed-Solomon codes with different code … MDS self-dual codes have nice algebraic structures, theoretical significance and practical implications. In this paper, we present three classes of $q^2$-ary Hermitian self-dual (extended) generalized Reed-Solomon codes with different code locators. Combining the results in Ball et al. (Designs, Codes and Cryptography, 89: 811-821, 2021), we show that if the code locators do not contain zero, $q^2$-ary Hermitian self-dual (extended) GRS codes of length $\geq 2q\ (q>2)$ does not exist. Under certain conditions, we prove Conjecture 3.7 and Conjecture 3.13 proposed by Guo and Li et al. (IEEE Communications Letters, 25(4): 1062-1065, 2021).

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The hull of two classical propagation rules and their applications

2022-01-01

Yang Li , Shixin Zhu

In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria … In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for the resulting codes derived from these two propagation rules being self-dual, self-orthogonal, or linear complementary dual (LCD) codes are given. As an application, we construct some linear codes with prescribed hull dimensions, many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, and quaternary Hermitian FSD LCD codes. Some new even-like, odd-like, Euclidean and Hermitian self-orthogonal codes are also obtained. Many of {these} codes are also (almost) optimal according to the Database maintained by Markus Grassl. Our methods contribute positively to improve the lower bounds on the minimum distance of known LCD codes.

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On the b-symbol weights of linear codes for large b

2025-05-08

Dongmei Huang , Qunying Liao , Gaohua Tang +1 more

A generalization of the Tang-Ding binary cyclic codes

2025-01-21

Ling Li , Minjia Shi , Sihui Tao +4 more

New ternary self-orthogonal codes and related LCD codes from weakly regular plateaued functions

2025-01-01

Dengcheng Xie , Shixin Zhu , Yang Li

New record-breaking binary linear codes constructed from group codes

2024-12-19

C. X. Yu , Shixin Zhu , Hao Chen +2 more

In this paper, we employ group rings and automorphism groups of binary linear codes to construct new record-breaking binary linear codes. We consider the semidirect product of abelian groups and … In this paper, we employ group rings and automorphism groups of binary linear codes to construct new record-breaking binary linear codes. We consider the semidirect product of abelian groups and cyclic groups and use these groups to construct linear codes. Finally, we obtain some linear codes which have better parameters than the code in \cite{bib5}. All the calculation results and corresponding data are listed in the paper or posted online.

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Galois self-orthogonal MDS codes with large dimensions

2024-12-06

Ruhao Wan , Shixin Zhu

Let $q=p^m$ be a prime power, $e$ be an integer with $0\leq e\leq m-1$ and $s=\gcd(e,m)$. In this paper, for a vector $v$ and a $q$-ary linear code $C$, we … Let $q=p^m$ be a prime power, $e$ be an integer with $0\leq e\leq m-1$ and $s=\gcd(e,m)$. In this paper, for a vector $v$ and a $q$-ary linear code $C$, we give some necessary and sufficient conditions for the equivalent code $vC$ of $C$ and the extended code of $vC$ to be $e$-Galois self-orthogonal. From this, we directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be $e$-Galois self-orthogonal. Furthermore, for all possible $e$ satisfying $0\leq e\leq m-1$, we classify them into three cases (1) $\frac{m}{s}$ odd and $p$ even; (2) $\frac{m}{s}$ odd and $p$ odd; (3) $\frac{m}{s}$ even, and construct several new classes of $e$-Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our $e$-Galois self-orthogonal MDS codes can have dimensions greater than $\lfloor \frac{n+p^e-1}{p^e+1}\rfloor$, which are not covered by previously known ones. Moreover, by propagation rules, we obtain some new MDS codes with Galois hulls of arbitrary dimensions. As an application, many quantum codes can be obtained from these MDS codes with Galois hulls.

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On $\sigma$ self-orthogonal matrix-product codes associated with Toeplitz matrices

2024-05-10

Yang Li , Shixin Zhu , Edgar Martı́nez-Moro

In this paper, we present four general constructions of $\sigma$ self-orthogonal matrix-product codes associated with Toeplitz matrices. The first one relies on the $\sigma'$ dual of a known $\sigma'$ dual-containing … In this paper, we present four general constructions of $\sigma$ self-orthogonal matrix-product codes associated with Toeplitz matrices. The first one relies on the $\sigma'$ dual of a known $\sigma'$ dual-containing matrix-product code; the second one is founded on quasi-$\widehat{\sigma}$ matrices, where we provide an efficient algorithm for generating them on the basic of Toeplitz matrices; and the last two ones are based on the utilization of certain special Toeplitz matrices. Concrete examples and detailed comparisons are provided. As a byproduct, we also find an application of Toeplitz matrices in $\widetilde{\tau}$-optimal defining matrices.

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Binary duadic codes and their related codes with a square-root-like lower bound

2024-04-27

Tingting Wu , Lanqiang Li , Xiuyu Zhang +1 more

Binary cyclic codes have been a hot topic for many years, and significant progress has been made in the study of this types of codes. As is well known, it … Binary cyclic codes have been a hot topic for many years, and significant progress has been made in the study of this types of codes. As is well known, it is hard to construct infinite families of binary cyclic codes [n, n+1/2] with good minimum distance. In this paper, by using the BCH bound on cyclic codes, one of the open problems proposed by Liu et al. about binary cyclic codes (Finite Field Appl 91:102270, 2023) is settled. Specially, we present several families of binary duadic codes with length 2^m-1 and dimension 2^(m-1), and the minimum distances have a square-root-like lower bound. As a by-product, the parameters of their dual codes and extended codes are provided, where the latter are self-dual and doubly-even.

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Optimal quinary cyclic codes with three zeros

2024-02-17

Tingting Wu , Shixin Zhu , Li Liu +1 more

Several new classes of MDS symbol-pair codes derived from matrix-product codes

2024-01-01

Xiujing Zheng Liqi Wang , Shixin Zhu

In order to correct the pair-errors generated during the transmission of modern high-density data storage that the outputs of the channels consist of overlapping pairs of symbols, a new coding … In order to correct the pair-errors generated during the transmission of modern high-density data storage that the outputs of the channels consist of overlapping pairs of symbols, a new coding scheme named symbol-pair code is proposed. The error-correcting capability of the symbol-pair code is determined by its minimum symbol-pair distance. For such codes, the larger the minimum symbol-pair distance, the better. It is a challenging task to construct symbol-pair codes with optimal parameters, especially, maximum-distance-separable (MDS) symbol-pair codes. In this paper, the permutation equivalence codes of matrix-product codes with underlying matrixes of orders 3 and 4 are used to extend the minimum symbol-pair distance, and six new classes of MDS symbol-pair codes are derived.

New non-GRS type MDS codes and NMDS codes

2024-01-01

Yang Li , Shixin Zhu

In this paper, we study a class of special linear codes involving their parameters, weight distributions, and self-orthogonal properties. On one hand, we prove that such codes must be maximum … In this paper, we study a class of special linear codes involving their parameters, weight distributions, and self-orthogonal properties. On one hand, we prove that such codes must be maximum distance separable (MDS) or near MDS (NMDS) codes and completely determine their weight distributions with the help of the solutions to some subset sum problems. Based on the well-known Schur method, we also show that such codes are non-equivalent to generalized Reed-Solomon codes. On the other hand, a sufficient and necessary condition for such codes to be self-orthogonal is characterized. Based on this condition, we further deduce that there are no self-dual codes in this class of linear codes and explicitly construct two classes of almost self-dual codes.

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Three classes of propagation rules for GRS and EGRS codes and their applications to EAQECCs

2024-01-01

Ruhao Wan , Shixin Zhu

In this paper, we study the Hermitian hulls of (extended) generalized Reed-Solomon (GRS and EGRS) codes over finite fields. For a given class of (extended) GRS codes, by increasing the … In this paper, we study the Hermitian hulls of (extended) generalized Reed-Solomon (GRS and EGRS) codes over finite fields. For a given class of (extended) GRS codes, by increasing the length, increasing the dimensions and increasing both the length and the dimensions, we obtain three new classes of (extended) GRS codes with Hermitian hulls of arbitrary dimensions. Furthermore, we obtain several new classes of $q^2$-ary maximum distance separable (MDS) codes with Hermitian hulls of arbitrary dimensions. And the dimension of these MDS codes can be taken from $1$ to $\frac{n}{2}$. By propagation rules, the parameters of the obtained code can be more flexible. As an application, a lot of new (MDS) entanglement-assisted quantum error correction codes (EAQECCs) can be constructed from previous known (extended) GRS codes. We derive three new propagation rules on (MDS) EAQECCs constructed from (extended) GRS codes. Finally, we present several new classes of (MDS) EAQECCs with flexible parameters. Notably, the distance parameters of our codes can range from $2$ to $\frac{n+2}{2}$.

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BCH codes with larger dimensional hull

2023-08-17

Binbin Pang , Shixin Zhu , Tian Yang +1 more

Several classes of Galois self-orthogonal MDS codes and related applications

2023-07-27

Yang Li , Yunfei Su , Shixin Zhu +2 more

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The Hull of Two Classical Propagation Rules and Their Applications

2023-06-20

Yang Li , Shixin Zhu , Edgar Martı́nez-Moro

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MDS codes with Euclidean and Hermitian hulls of flexible dimensions and their applications to EAQECCs

2023-03-22

Yang Li , Ruhao Wan , Shixin Zhu

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New Quantum MDS codes from Hermitian self-orthogonal generalized Reed-Solomon codes

2023-01-01

Ruhao Wan , Shixin Zhu

Quantum maximum-distance-separable (MDS for short) codes are an important class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed-Solomon (GRS for short) codes, we construct five new … Quantum maximum-distance-separable (MDS for short) codes are an important class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed-Solomon (GRS for short) codes, we construct five new classes of $q$-ary quantum MDS codes with minimum distance larger than $q/2+1$. Furthermore, the parameters of our quantum MDS code cannot be obtained from the previous constructions.

New entanglement-assisted quantum codes from negacyclic codes

2023-01-01

Xiaojing Chen , Xingbo Lu , Shixin Zhu +2 more

The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the … The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the duality condition and utilizing preshared entanglement between the sender and receiver. In this paper, a new family of EAQECCs is constructed from negacyclic codes of length $n=\frac{q^2+1}{a}$, where $q$ is an odd prime power, $a=\frac{m^2+1}{2}$ and $m$ is an odd integer. Some new entanglement-assisted quantum maximum distance separable (EAQMDS) codes are obtained in the sense that their parameters are not covered by the previously known ones.

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Symplectic self-orthogonal and LCD codes from the Plotkin sum construction

2023-01-01

Shitao Li , Yang Li , Minjia Shi +1 more

In this work, we propose two criteria for linear codes obtained from the Plotkin sum construction being symplectic self-orthogonal (SO) and linear complementary dual (LCD). As specific constructions, several classes … In this work, we propose two criteria for linear codes obtained from the Plotkin sum construction being symplectic self-orthogonal (SO) and linear complementary dual (LCD). As specific constructions, several classes of symplectic SO codes with good parameters including symplectic maximum distance separable codes are derived via $\ell$-intersection pairs of linear codes and generalized Reed-Muller codes. Also symplectic LCD codes are constructed from general linear codes. Furthermore, we obtain some binary symplectic LCD codes, which are equivalent to quaternary trace Hermitian additive complementary dual codes that outperform best-known quaternary Hermitian LCD codes reported in the literature. In addition, we prove that symplectic SO and LCD codes obtained in these ways are asymptotically good.

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On $\ell$-MDS codes and a conjecture on infinite families of $1$-MDS codes

2023-01-01

Yang Li , Shixin Zhu , Edgar Martı́nez-Moro

The class of $\ell$-maximum distance separable ($\ell$-MDS) codes {is a} generalization of maximum distance separable (MDS) codes {that} has attracted a lot of attention due to its applications in several … The class of $\ell$-maximum distance separable ($\ell$-MDS) codes {is a} generalization of maximum distance separable (MDS) codes {that} has attracted a lot of attention due to its applications in several areas such as secret sharing schemes, index coding problems, informed source coding problems, and combinatorial $t$-designs. In this paper, for $\ell=1$, we completely solve a conjecture recently proposed by Heng $et~al.$ (Discrete Mathematics, 346(10): 113538, 2023) and obtain infinite families of $1$-MDS codes with general dimensions holding $2$-designs. These later codes are also been proven to be optimal locally recoverable codes. For general {positive integers} $\ell$ and $\ell'$, we construct new $\ell$-MDS codes from known $\ell'$-MDS codes via some classical propagation rules involving the extended, expurgated, and $(u,u+v)$ constructions. Finally, we study some general results including characterization, weight distributions, and bounds on maximum lengths of $\ell$-MDS codes, which generalize, simplify, or improve some known results in the literature.

A Generalization of the Tang-Ding Binary Cyclic Codes

2023-01-01

Zhonghua Sun , Ling Li , Shixin Zhu

Cyclic codes are an interesting family of linear codes since they have efficient decoding algorithms and contain optimal codes as subfamilies. Constructing infinite families of cyclic codes with good parameters … Cyclic codes are an interesting family of linear codes since they have efficient decoding algorithms and contain optimal codes as subfamilies. Constructing infinite families of cyclic codes with good parameters is important in both theory and practice. Recently, Tang and Ding [IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842--7849, 2022] proposed an infinite family of binary cyclic codes with good parameters. Shi et al. [arXiv:2309.12003v1, 2023] developed the binary Tang-Ding codes to the $4$-ary case. Inspired by these two works, we study $2^s$-ary Tang-Ding codes, where $s\geq 2$. Good lower bounds on the minimum distance of the $2^s$-ary Tang-Ding codes are presented. As a by-product, an infinite family of $2^s$-ary duadic codes with a square-root like lower bound is presented.

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New ternary self-orthogonal codes and related LCD codes from weakly regular plateaued functions

2023-01-01

Dengcheng Xie , Shixin Zhu , Yang Li

A linear code is said to be self-orthogonal if it is contained in its dual. Self-orthogonal codes are of interest because of their important applications, such as for constructing linear … A linear code is said to be self-orthogonal if it is contained in its dual. Self-orthogonal codes are of interest because of their important applications, such as for constructing linear complementary dual (LCD) codes and quantum codes. In this paper, we construct several new families of ternary self-orthogonal codes by employing weakly regular plateaued functions. Their parameters and weight distributions are completely determined. Then we apply these self-orthogonal codes to construct several new families of ternary LCD codes. As a consequence, we obtain many (almost) optimal ternary self-orthogonal codes and LCD codes.

On MDS codes with galois hulls of arbitrary dimensions

2022-12-17

Yang Li , Shixin Zhu , Ping Li

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On MDS Codes With Galois Hulls of Arbitrary Dimensions

2022-09-13

Yang Li , Shixin Zhu , Ping Li

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Construction of self-dual MDR cyclic codes over finite chain rings

2022-06-12

Jian Yuan , Shixin Zhu , Xiaoshan Kai

New MDS self-dual codes over finite fields $\F_{r^2}$

2022-01-01

Ruhao Wan , Shixin Zhu

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Construction of MDS self-dual codes from generalized Reed-Solomon codes

2022-01-01

Ruhao Wan , Shixin Zhu , Jin Li

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On Galois hulls of linear codes and new entanglement-assisted quantum error-correcting codes

2022-01-01

Yang Li , Shixin Zhu

Several classes of Galois self-orthogonal MDS codes and related applications

2022-01-01

Yang Li , Yunfei Su , Shixin Zhu

MDS Codes with Euclidean and Hermitian Hulls of Flexible Dimensions and Their Applications to EAQECCs

2022-01-01

Yang Li , Ruhao Wan , Shixin Zhu

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Research on Hermitian self-dual codes, GRS codes and EGRS codes

2022-01-01

Ruhao Wan , Shixin Zhu

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The hull of two classical propagation rules and their applications

2022-01-01

Yang Li , Shixin Zhu

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Repeated-root constacyclic codes of length 6lmpn

2021-09-27

Tingting Wu , Shixin Zhu , Li Liu +1 more

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Cyclic codes and some new entanglement-assisted quantum MDS codes

2021-09-06

Xiaojing Chen , Shixin Zhu , Wan Jiang

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New entanglement-assisted quantum MDS codes with length $$n=\frac{q^2+1}{5}$$

2020-06-10

Shixin Zhu , Wan Jiang , Xiaojing Chen

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Some new bounds on LCD codes over finite fields

2020-01-09

Binbin Pang , Shixin Zhu , Xiaoshan Kai

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On the constructions of $n$-cycle permutations

2020-01-01

Yuting Chen , Liqi Wang , Shixin Zhu

Cyclic codes and some new entanglement-assisted quantum MDS codes

2020-01-01

Wan Jiang , Shixin Zhu , Xiaojing Chen

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Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

2019-10-30

Weijun Fang , Fang‐Wei Fu , Lanqiang Li +1 more

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On Self‐dual and LCD Double Circulant Codes over a Non‐chain Ring*

2019-09-01

Ting Yao , Shixin Zhu , Xiaoshan Kai

A Class of Optimal Cyclic Codes With Two Zeros

2019-06-06

Dengchuan Liao , Xiaoshan Kai , Shixin Zhu +1 more

A Class of Narrow-Sense BCH Codes

2019-05-24

Shixin Zhu , Zhonghua Sun , Xiaoshan Kai

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A Class of Narrow-Sense BCH Codes

2019-01-01

Shixin Zhu , Zhonghua Sun , Xiaoshan Kai

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New entanglement-assisted quantum MDS codes with length $n=\frac{q^2+1}5$

2019-01-01

Shixin Zhu , Wan Jiang , Xiaojing Chen

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MDS codes with Hermitian hulls of arbitrary dimensions and their quantum error correction.

2018-12-21

Lanqiang Li , Shixin Zhu , Li Liu +1 more

Entanglement-assisted quantum MDS codes constructed from constacyclic codes

2018-09-06

Xiaojing Chen , Shixin Zhu , Xiaoshan Kai

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Entanglement-assisted quantum MDS codes constructed from constacyclic codes

2018-01-01

Xiaojing Chen , Shixin Zhu , Xiaoshan Kai

On cyclic codes of length $2^e$ over finite fields

2018-01-01

Binbin Pang , Shixin Zhu , Ping Li

A family of constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ and application to quantum codes

2017-12-06

Yongsheng Tang , Ting Yao , Shixin Zhu +1 more

On LCD Negacyclic Codes over Finite Fields

2017-11-29

Binbin Pang , Shixin Zhu , Zhonghua Sun

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Several classes of optimal ternary cyclic codes

2017-01-01

Lanqiang Li , Li Liu , Shixin Zhu

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Coauthor	Papers Together
Xiaoshan Kai	15
Xiaojing Chen	8
Yang Li	7
Ruhao Wan	7
Lanqiang Li	7
Wan Jiang	5
Li Liu	5
Zhonghua Sun	5
Liqi Wang	4
Yongsheng Tang	4
Yang Li	4
Binbin Pang	4
Minjia Shi	4
Edgar Martı́nez-Moro	3
Yang Li	3
Ting Yao	3
Yunfei Su	2
Ping Li	2
Tingting Wu	2
Jian Ding	2
Xiuyu Zhang	2
Ping Li	2
Shitao Li	2
Dengcheng Xie	2
Bennian Dou	1
Tingting Wu	1
Liqi Wang	1
Dengchuan Liao	1
Hao Chen	1
Jin Li	1
Li Liu	1
Jon-Lark Kim	1
Chungen Xu	1
Patrick Solé	1
Yang Li	1
Ling Li	1
Yuting Chen	1
Gaohua Tang	1
Yang Li	1
Sihui Tao	1
Zhonghua Sun	1
Li Ping	1
Ling Li	1
Jun Gao	1
C. X. Yu	1
Jian Yuan	1
Keqin Feng	1
Xingbo Lu	1
Dongmei Huang	1
Minghui Yang	1

Good quantum error-correcting codes exist

1996-08-01

A.R. Calderbank , Peter W. Shor

A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such … A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n = 1 - 2H(2t/n) where H(p) is the binary entropy function -p log p - (1-p) log (1-p). Upper bounds on this asymptotic rate are given.

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Constructions of good entanglement-assisted quantum error correcting codes

2017-01-25

Kenza Guenda , Somphong Jitman , T. Aaron Gulliver

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LCD Cyclic Codes Over Finite Fields

2017-02-22

Chengju Li , Cunsheng Ding , Shuxing Li

In addition to their applications in data storage, communications systems, and consumer electronics, linear complementary dual (LCD) codes-a class of linear codes-have been employed in cryptography recently. LCD cyclic codes … In addition to their applications in data storage, communications systems, and consumer electronics, linear complementary dual (LCD) codes-a class of linear codes-have been employed in cryptography recently. LCD cyclic codes were referred to as reversible cyclic codes in the literature. The objective of this paper is to construct several families of reversible cyclic codes over finite fields and analyze their parameters. The LCD cyclic codes presented in this paper have very good parameters in general, and contain many optimal codes. A well rounded treatment of reversible cyclic codes is also given in this paper.

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Correcting Quantum Errors with Entanglement

2006-09-29

Todd A. Brun , Igor Devetak , Min-Hsiu Hsieh

We show how entanglement shared between encoder and decoder can simplify the theory of quantum error correction. The entanglement-assisted quantum codes we describe do not require the dual-containing constraint necessary … We show how entanglement shared between encoder and decoder can simplify the theory of quantum error correction. The entanglement-assisted quantum codes we describe do not require the dual-containing constraint necessary for standard quantum error-correcting codes, thus allowing us to "quantize" all of classical linear coding theory. In particular, efficient modern classical codes that attain the Shannon capacity can be made into entanglement-assisted quantum codes attaining the hashing bound (closely related to the quantum capacity). For systems without large amounts of shared entanglement, these codes can also be used as catalytic codes, in which a small amount of initial entanglement enables quantum communication.

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On Quantum and Classical BCH Codes

2007-03-01

Salah A. Aly , Andreas Klappenecker , Pradeep Kiran Sarvepalli

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Classical Bose–Chaudhuri–Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such … <para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Classical Bose–Chaudhuri–Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length <emphasis><formula formulatype="inline"> <tex>$n$</tex></formula></emphasis> can contain its dual code only if its designed distance <emphasis><formula formulatype="inline"><tex>$\delta =O(\sqrt {n})$</tex></formula></emphasis>, and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and – consequently – the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters. </para>

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Entanglement-assisted quantum MDS codes constructed from constacyclic codes

2018-09-06

Xiaojing Chen , Shixin Zhu , Xiaoshan Kai

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Computing automorphism groups of error-correcting codes

1982-05-01

Jeffrey S. Leon

An algorithm is described for computing the automorphism group of an error correcting code. The algorithm determines the order of the automorphism group and produces a set of monomial permutations … An algorithm is described for computing the automorphism group of an error correcting code. The algorithm determines the order of the automorphism group and produces a set of monomial permutations which generate the group. It has been implemented on a computer and has been used successfully on a great number of codes of moderate length.

Nonbinary quantum stabilizer codes

2001-01-01

Alexei Ashikhmin , Emanuel Knill

We define and show how to construct nonbinary quantum stabilizer codes. Our approach is based on nonbinary error bases. It generalizes the relationship between self-orthogonal codes over F/sub 4/ and … We define and show how to construct nonbinary quantum stabilizer codes. Our approach is based on nonbinary error bases. It generalizes the relationship between self-orthogonal codes over F/sub 4/ and binary quantum codes to one between self-orthogonal codes over F(q/sup 2/) and q-ary quantum codes for any prime power q.

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Cyclic and Negacyclic Codes Over Finite Chain Rings

2004-07-27

Hai Q. Dinh , Sergio R. López-Permouth

The structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring R are established when n is not divisible by the characteristic of … The structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring R are established when n is not divisible by the characteristic of the residue field R~. Some cases where n is divisible by the characteristic of the residue field R~ are also considered. Namely, the structure of negacyclic codes of length 2/sup t/ over /spl Zopf//sub 2//sup m/ and that of their duals are derived.

Permutation group algorithms based on partitions, I: Theory and algorithms

1991-10-01

Jeffrey S. Leon

Entanglement-assisted quantum communication beating the quantum Singleton bound

2021-02-24

Markus Grassl

Brun, Devetak, and Hsieh [Science 314, 436 (2006)] demonstrated that preshared entanglement between the sender and receiver enables quantum communication protocols that have better parameters than schemes without the assistance … Brun, Devetak, and Hsieh [Science 314, 436 (2006)] demonstrated that preshared entanglement between the sender and receiver enables quantum communication protocols that have better parameters than schemes without the assistance of entanglement. Subsequently, the same authors derived a version of the so-called quantum Singleton bound that relates the parameters of the entanglement-assisted quantum-error-correcting codes proposed by them. We present an entanglement-assisted quantum communication scheme with parameters violating this bound in certain ranges. For a fixed transmission rate, our scheme allows one to correct a larger fraction of errors.

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Linear Codes Over $\mathbb F_q$ Are Equivalent to LCD Codes for $q>3$

2018-01-04

Claude Carlet , Sihem Mesnager , Chunming Tang +2 more

The hull $H(C)$ of a linear code $C$ is defined by $H(C)=C \cap C^\perp$. A linear code with a complementary dual (LCD) is a linear code with $H(C)=\{0\}$. The dimension … The hull $H(C)$ of a linear code $C$ is defined by $H(C)=C \cap C^\perp$. A linear code with a complementary dual (LCD) is a linear code with $H(C)=\{0\}$. The dimension of the hull of a code is an invariant under permutation equivalence. For binary and ternary codes the dimension of the hull is also invariant under monomial equivalence and we show that this invariant is determined by the extended weight enumerator of the code.\\ The hull of a code is not invariant under monomial equivalence if $q\geq 4$. We show that every ${\mathbb F}_q $-linear code is monomial equivalent with an LCD code in case $q \geq 4$. The proof uses techniques from Gr\"obner basis theory. We conclude that if there exists an ${\mathbb F}_q $-linear code with parameters $[n,k,d]_q$ and $q \geq 4$, then there exists also a LCD code with the same parameters. Hence this holds for optimal and MDS codes. In particular there exist LCD codes that are above the Gilbert-Varshamov bound if $q$ is a square and $q\geq 49$ by the existence of such codes that are algebraic geometric.\\ Similar results are obtained with respect to Hermitian LCD codes.

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Constructions of q-ary entanglement-assisted quantum MDS codes with minimum distance greater than q+1

2016-04-01

Jihao Fan , Hanwu Chen , Juan Xu

he entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using … he entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum MDS (QMDS) codes of the same length, and three classes of these EAQMDS codes consume only one pair of maximally entangled states.

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Two new classes of quantum MDS codes

2018-06-19

Weijun Fang , Fang‐Wei Fu

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Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

2019-10-30

Weijun Fang , Fang‐Wei Fu , Lanqiang Li +1 more

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Nonbinary Stabilizer Codes Over Finite Fields

2006-10-30

Avanti Ulhas Ketkar , Andreas Klappenecker , Santhosh Kumar +1 more

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have … One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$bf F_q$</tex> in terms of classical codes over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$ bf F_q^2$</tex> is provided that generalizes the well-known notion of additive codes over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$ bf F_4$</tex> of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose–Chaudhuri–Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

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The Magma Algebra System I: The User Language

1997-09-01

Wieb Bosma , John Cannon , Catherine Playoust

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Galois self-dual constacyclic codes

2016-09-29

Yun Fan , Liang Zhang

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Optimal entanglement formulas for entanglement-assisted quantum coding

2008-06-19

Mark M. Wilde , Todd A. Brun

We provide several formulas that determine the optimal number of entangled bits (ebits) that a general entanglement-assisted quantum code requires. Our first theorem gives a formula that applies to an … We provide several formulas that determine the optimal number of entangled bits (ebits) that a general entanglement-assisted quantum code requires. Our first theorem gives a formula that applies to an arbitrary entanglement-assisted block code. Corollaries of this theorem give formulas that apply to a code imported from two classical binary block codes, to a code imported from a classical quaternary block code, and to a continuous-variable entanglement-assisted quantum block code. Finally, we conjecture two formulas that apply to entanglement-assisted quantum convolutional codes.

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Affine and projective planes

1990-08-01

E. F. Assmus , J. D. Key

ON OPTIMAL QUANTUM CODES

2004-03-01

Markus Grassl , Thomas Beth , Martin Rötteler

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum … We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛 q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q 2 , q 2 -2d+2, d〛 q for 1≤d≤q which additionally give rise to shortened codes 〚q 2 -s, q 2 -2d+2-s, d〛 q for some s.

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Galois hulls of linear codes over finite fields

2019-09-27

Hongwei Liu , Xu Pan

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Quantum generalized Reed-Solomon codes: Unified framework for quantum maximum-distance-separable codes

2008-01-09

Zhuo Li , Lijuan Xing , Xinmei Wang

We construct a family of quantum maximum-distance-separable (MDS) codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give … We construct a family of quantum maximum-distance-separable (MDS) codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give code bounds and show how to construct them analytically. We find that existing quantum MDS codes can be unified under these codes in the sense that when a quantum MDS code exists, then a quantum code of this type with the same parameters also exists. Thus, as far as is known at present, they are the most important family of quantum MDS codes.

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Optimal Ternary Cyclic Codes From Monomials

2013-05-01

Cunsheng Ding , Tor Helleseth

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Perfect nonlinear … Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters [3 m -1, 3 m -1-2m, 4] by Carlet, Ding, and Yuan in 2005. In this paper, almost perfect nonlinear monomials, and a number of other monomials over GF(3 m ) are used to construct optimal ternary cyclic codes with the same parameters. Nine open problems on such codes are also presented.

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Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance

2018-07-20

Liangdong Lü , Wenping Ma , Ruihu Li +3 more

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A class of three-weight cyclic codes

2013-09-17

Zhengchun Zhou , Cunsheng Ding

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New quantum MDS codes derived from constacyclic codes

2015-01-06

Liqi Wang , Shixin Zhu

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New Constructions of MDS Codes With Complementary Duals

2017-09-04

Bocong Chen , Hongwei Liu

Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD … Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for given length and code size. Constructing LCD MDS codes is thus of significance in theory and practice. Recently, Jin constructed several classes of LCD MDS codes through generalized Reed-Solomon codes. In this paper, a different approach is proposed to obtain new LCD MDS codes from generalized Reed-Solomon codes. Consequently, new code constructions are provided and certain previously known results by Jin are extended.

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Euclidean and Hermitian LCD MDS codes

2018-02-05

Claude Carlet , Sihem Mesnager , Chunming Tang +1 more

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Optimal p-ary cyclic codes with minimum distance four from monomials

2015-10-03

Guangkui Xu , Xiwang Cao , Shanding Xu

Quasi-cyclic complementary dual codes

2016-07-23

Cem Güneri̇ , Buket Özkaya , Patrick Solé

On Self-Dual Cyclic Codes Over Finite Fields

2011-03-15

Yan Jia , San Ling , Chaoping Xing

In coding theory, self-dual codes and cyclic codes are important classes of codes which have been extensively studied. The main objects of study in this paper are self-dual cyclic codes … In coding theory, self-dual codes and cyclic codes are important classes of codes which have been extensively studied. The main objects of study in this paper are self-dual cyclic codes over finite fields, i.e., the intersection of these two classes. We show that self-dual cyclic codes of length n over \BBF q exist if and only if n is even and q = 2 m with m a positive integer. The enumeration of such codes is also investigated. When n and q are even, there is always a trivial self-dual cyclic code with generator polynomial x n / 2 +1. We, therefore, classify the existence of self-dual cyclic codes, for given n and q , into two cases: when only the trivial one exists and when two or more such codes exist. Given n and m , an easy criterion to determine which of these two cases occurs is given in terms of the prime factors of n , for most n . We also show that, over a fixed field, the latter case occurs more frequently as the length grows.

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Dimensions of three types of BCH codes over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mtext>GF</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>

2017-04-24

Hao Liu , Cunsheng Ding , Chengju Li

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New Galois Hulls of GRS Codes and Application to EAQECCs

2021-08-27

Xiaolei Fang , Renjie Jin , Jinquan Luo +1 more

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Elementary Number Theory

1979-09-01

Henry J. Ricardo , David Burton

1. Some Preliminary Considerations. 2. Divisibility Theory in the Integers. 3. Primes and Their Distribution. 4. The Theory of Congruences. 5. Fermat's Theorem. 6. Number-Theoretic Functions. 7. Euler's Generalization of … 1. Some Preliminary Considerations. 2. Divisibility Theory in the Integers. 3. Primes and Their Distribution. 4. The Theory of Congruences. 5. Fermat's Theorem. 6. Number-Theoretic Functions. 7. Euler's Generalization of Fermat's Theorem. 8. Primitive Roots and Indices. 9. The Quadratic Reciprocity Law. 10. Perfect Numbers. 11. The Fermat Conjecture. 12. Representation of Integers as Sums of Squares. 13. Fibonacci Numbers. 14. Continued Fractions. 15. Some Twentieth-Century Developments.

Optimal ternary cyclic codes with minimum distance four and five

2014-07-15

Nian Li , Chunlei Li , Tor Helleseth +2 more

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Graphical nonbinary quantum error-correcting codes

2008-07-03

Dan Hu , Weidong Tang , Meisheng Zhao +3 more

In this paper, we present a systematic way based on the nonbinary graph state of constructing good nonbinary quantum codes, both additive and nonadditive, for systems with integer dimensions. With … In this paper, we present a systematic way based on the nonbinary graph state of constructing good nonbinary quantum codes, both additive and nonadditive, for systems with integer dimensions. With a computer search, which results in many interesting codes including some nonadditive codes meeting the Singleton bounds, we are able to construct explicitly four families of optimal codes, namely, ${[[6,2,3]]}_{p}$, ${[[7,3,3]]}_{p}$, ${[[8,2,4]]}_{p}$, and ${[[8,4,3]]}_{p}$ for any odd dimension $\mathit{p}$ and a family of nonadditive codes ${((5,p,3))}_{p}$ for arbitrary $p>3$. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes ${((6,2{p}^{2},3))}_{2p}$ for odd $\mathit{p}$, whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem.

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Skew multi-twisted codes over finite fields and their Galois duals

2019-07-11

Anuradha Sharma , Varsha Chauhan

Theory of quantum error-correcting codes

1997-02-01

Emanuel Knill , Raymond Laflamme

Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction … Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.

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Some bounds on binary LCD codes

2017-09-26

Lucky Galvez , Jon-Lark Kim , Nari Lee +2 more

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Algebraic Geometry Codes With Complementary Duals Exceed the Asymptotic Gilbert-Varshamov Bound

2017-11-13

Lingfei Jin , Chaoping Xing

It was shown by Massey that linear complementary dual (LCD) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV) bound. Until now, the … It was shown by Massey that linear complementary dual (LCD) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound.

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Dimensions of nonbinary antiprimitive BCH codes and some conjectures

2017-01-01

Yang Liu , Ruihu Li , Luobin Guo +1 more

Bose-Chaudhuri-Hocquenghem (BCH) codes have been intensively investigated. Even so, there is only a little known about primitive BCH codes, let alone non-primitive ones. In this paper, let $q>2$ be a … Bose-Chaudhuri-Hocquenghem (BCH) codes have been intensively investigated. Even so, there is only a little known about primitive BCH codes, let alone non-primitive ones. In this paper, let $q>2$ be a prime power, the dimension of a family of non-primitive BCH codes of length $n=q^{m}+1$ (also called antiprimitive) is studied. These codes are also linear codes with complementary duals (called LCD codes). Through some approaches such as iterative algorithm, partition and scaling, all coset leaders of $C_{x}$ modulo $n$ with $q^{\lceil \frac{m}{2}\rceil}

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Complementary Dual Algebraic Geometry Codes

2017-10-24

Sihem Mesnager , Chunming Tang , Yanfeng Qi

Linear complementary dual (LCD) codes are a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications … Linear complementary dual (LCD) codes are a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes have been employed in cryptography. More specifically, it has been shown that LCD codes can also help improve the security of the information processed by sensitive devices, especially against so-called sidechannel attacks (SCA) and fault non-invasive attacks. In this paper, we are interested in the construction of particular algebraic geometry LCD codes which could be good candidates to be resistant against SCA. We firstly provide a construction scheme for obtaining LCD codes from any algebraic curve. Then, some explicit LCD codes from elliptic curves are presented. Maximum distance separable (MDS) codes are of the most importance in coding theory due to their theoretical significance and practical interests. In this paper, all the constructed LCD codes from elliptic curves are MDS or almost MDS. Some infinite classes of LCD codes from elliptic curves are optimal due to the Griesmer bound. Finally, we also derive some explicit LCD codes from hyperelliptic curves and Hermitian curves.

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Modular andp-adic cyclic codes

1995-07-01

A.R. Calderbank , N. J. A. Sloane

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Are&lt;tex&gt;$2$&lt;/tex&gt;-Weight Projective Cyclic Codes Irreducible?

2005-01-24

J. Wolfmann

We prove the following result on a 2-weight projective cyclic code C over F q . If q=2 then C is irreducible. If qne2 then either C is … We prove the following result on a 2-weight projective cyclic code C over F q . If q=2 then C is irreducible. If qne2 then either C is irreducible or C is the direct sum of two 1-weight irreducible cyclic codes

Simple quantum error-correcting codes

1996-12-01

Andrew Steane

Methods of finding good quantum error-correcting codes are discussed, and many example codes are presented. The recipe ${\mathit{C}}_{2}^{\mathrm{\ensuremath{\perp}}}$\ensuremath{\subset}${\mathit{C}}_{1}$, where ${\mathit{C}}_{1}$ and ${\mathit{C}}_{2}$ are classical codes, is used to obtain codes … Methods of finding good quantum error-correcting codes are discussed, and many example codes are presented. The recipe ${\mathit{C}}_{2}^{\mathrm{\ensuremath{\perp}}}$\ensuremath{\subset}${\mathit{C}}_{1}$, where ${\mathit{C}}_{1}$ and ${\mathit{C}}_{2}$ are classical codes, is used to obtain codes for up to 16 information quantum bits (qubits) with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing ${\mathit{C}}_{1}$ to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to single-error-correcting codes for 3, 4, and 5 information qubits with block lengths of 8, 10, and 11 qubits, respectively. \textcopyright{} 1996 The American Physical Society.

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The dimension and minimum distance of two classes of primitive BCH codes

2017-01-10

Cunsheng Ding , Cuiling Fan , Zhengchun Zhou

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Construction of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:msub><mml:mrow /><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>quantum codes for odd prime power<mml:…

2010-11-16

Ruihu Li , Zongben Xu

For each odd prime power $q$, let $4 \leq n\leq q^{2}+1$. Hermitian self-orthogonal $[n,2,n-1]$ codes over $GF(q^{2})$ with dual distance three are constructed by using finite field theory. Hence, $[[n,n-4,3]]_{q}$ … For each odd prime power $q$, let $4 \leq n\leq q^{2}+1$. Hermitian self-orthogonal $[n,2,n-1]$ codes over $GF(q^{2})$ with dual distance three are constructed by using finite field theory. Hence, $[[n,n-4,3]]_{q}$ quantum MDS codes for $4 \leq n\leq q^{2}+1$ are obtained.

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Repeated-root constacyclic codes of length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>ℓ</mml:mi><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup></mml:math> and their duals

2014-06-16

Bocong Chen , Hai Q. Dinh , Hongwei Liu

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Duality in Entanglement-Assisted Quantum Error Correction

2013-02-08

Ching–Yi Lai , Todd A. Brun , Mark M. Wilde

The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is defined from the orthogonal group of a simplified stabilizer group. From the Poisson summation formula, this duality leads to the … The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is defined from the orthogonal group of a simplified stabilizer group. From the Poisson summation formula, this duality leads to the MacWilliams identities and linear programming bounds for EAQEC codes. We establish a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits.

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