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Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

2021-04-01
Chunming Tang , Yan Qiu , Qunying Liao +1 more
In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, recently introduced by Bonini and Borello, and then completely characterize minimal linear … In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension 3 and t-fold blocking sets with t ≥ 2 in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg's condition are obtained. In addition to this, open problems on the parameters and the weight distributions of some generated linear codes are presented.
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On the divisibility of power LCM matrices by power GCD matrices

2007-03-01
Jianrong Zhao , Shaofang Hong , Qunying Liao +1 more
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Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

2019-01-01
Chunming Tang , Yan Qiu , Qunying Liao +1 more
In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize … In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension $3$ and $t$-fold blocking sets with $t\ge 2$ in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. This confirms a recent conjecture by Alfarano, Borello and Neri in [a geometric characterization of minimal codes and their asymptotic performance, arXiv:1911.11738, 2019] about a lower bound of the minimum distance of a minimal code. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg's condition are obtained. In addition to this, the weight distributions of two subfamilies of the proposed minimal linear codes are established. Open problems are also presented.
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Complete weight enumerators for several classes of two-weight and three-weight linear codes

2021-07-29
Canze Zhu , Qunying Liao
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On an identity associated with Weil's estimate and its applications

2009-02-22
Yuan He , Qunying Liao

Some Congruences Involving Euler Numbers

2008-08-01
Yuan He , Qunying Liao
In this paper, we obtain some explicit congruences for Euler numbers modulo an odd prime power in an elementary way. The classical Bernoulli polynomials Bn(x) and Euler polynomials En(x) are … In this paper, we obtain some explicit congruences for Euler numbers modulo an odd prime power in an elementary way. The classical Bernoulli polynomials Bn(x) and Euler polynomials En(x) are usually deflned by the exponential generating functions: te xt e t i 1 = 1 X n=0 Bn(x) t n n! and 2e xt e t + 1 = 1 X n=0 En(x) t n n! :

New Deep Holes of Generalized Reed-Solomon Codes

2012-01-01
Jun Zhang , Fang‐Wei Fu , Qunying Liao
Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong \cite{WH} found a new class of deep holes for standard Reed-Solomon codes. In … Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong \cite{WH} found a new class of deep holes for standard Reed-Solomon codes. In the present paper, we give a concise method to obtain a new class of deep holes for generalized Reed-Solomon codes. In particular, for standard Reed-Solomon codes, we get the new class of deep holes given in \cite{WH}. Li and Wan \cite{L.W1} studied deep holes of generalized Reed-Solomon codes $GRS_{k}(\f,D)$ and characterized deep holes defined by polynomials of degree $k+1$. They showed that this problem is reduced to be a subset sum problem in finite fields. Using the method of Li and Wan, we obtain some new deep holes for special Reed-Solomon codes over finite fields with even characteristic. Furthermore, we study deep holes of the extended Reed-Solomon code, i.e., $D=\f$ and show polynomials of degree $k+2$ can not define deep holes.
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Two new classes of projective two-weight linear codes

2023-03-04
Canze Zhu , Qunying Liao
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A CLASS OF NEW NEAR-PERFECT NUMBERS

2015-06-01
Yanbin Li , Qunying Liao
Let <TEX>${\alpha}$</TEX> be a positive integer, and let <TEX>$p_1$</TEX>, <TEX>$p_2$</TEX> be two distinct prime numbers with <TEX>$p_1$</TEX> < <TEX>$p_2$</TEX>. By using elementary methods, we give two equivalent conditions of all … Let <TEX>${\alpha}$</TEX> be a positive integer, and let <TEX>$p_1$</TEX>, <TEX>$p_2$</TEX> be two distinct prime numbers with <TEX>$p_1$</TEX> < <TEX>$p_2$</TEX>. By using elementary methods, we give two equivalent conditions of all even near-perfect numbers in the form <TEX>$2^{\alpha}p_1p_2$</TEX> and <TEX>$2^{\alpha}p_1^2p_2$</TEX>, and obtain a lot of new near-perfect numbers which involve some special kinds of prime number pairs. One kind is exactly the new Mersenne conjecture's prime number pair. Another kind has the form <TEX>$p_1=2^{{\alpha}+1}-1$</TEX> and <TEX>$p_2={\frac{p^2_1+p_1+1}{3}}$</TEX>, where the former is a Mersenne prime and the latter's behavior is very much like a Fermat number.
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A class of double-twisted generalized Reed-Solomon codes

2024-03-01
Canze Zhu , Qunying Liao

On the Distribution of Normal Bases Over Finite Fields

2010-01-01
Qunying Liao

Self-dual twisted generalized Reed-Solomon codes

2021-01-01
Canze Zhu , Qunying Liao
In this paper, by using some properties for linear algebra methods, the parity-check matrices for twisted generalized Reed-Solomon codes with any given hook $h$ and twist $t$ are presented, and … In this paper, by using some properties for linear algebra methods, the parity-check matrices for twisted generalized Reed-Solomon codes with any given hook $h$ and twist $t$ are presented, and then a sufficient and necessary condition for the twisted generalized Reed-Solomon code with $h\ge t$ to be self-dual is given. Furthermore, several classes of self-dual codes with small Singleton defect are constructed based on twisted generalized Reed-Solomon codes, especially some of these self-dual codes are MDS or NMDS.
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The (+)-extended twisted generalized Reed-Solomon code

2023-10-17
Canze Zhu , Qunying Liao
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A recursion formula for the generalized Euler function $ φ_e(n) $

2021-01-01
Canze Zhu , Qunying Liao
In this paper, basing on the linear algebra methods and elementary techniques, for any positive integers $ e $ and $ n $, we obtain a recursion formula for the … In this paper, basing on the linear algebra methods and elementary techniques, for any positive integers $ e $ and $ n $, we obtain a recursion formula for the generalized Euler function $ \varphi_e(n) $, which is determined by some matrices related to a congruence equation modulo $ e $. Furthermore, through the recursion formula, we get the explicit formula for $ \varphi_5(n) $. Our results generalize the corresponding results in \cite{A4,A8,A10,A11}.
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Finding normal bases over finite fields with prescribed trace self-orthogonal relations

2014-02-17
Xiyong Zhang , Rongquan Feng , Qunying Liao +1 more
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The non-GRS properties for the twisted generalized Reed-Solomon code and its extended code

2022-01-01
Canze Zhu , Qunying Liao
In 2017, Beelen et al. firstly introduced twisted generalized Reed-Solomon (in short, TGRS) codes, and constructed a large subclass of MDS TGRS codes. Later, they proved that TGRS code is … In 2017, Beelen et al. firstly introduced twisted generalized Reed-Solomon (in short, TGRS) codes, and constructed a large subclass of MDS TGRS codes. Later, they proved that TGRS code is non-GRS when the code rate is less than one half. In this letter, basing on the dual code of the TGRS code or the extended TGRS code, by using the Schur product, we prove that almost all of TGRS codes and extended TGRS codes are non-GRS when the code rate more than one half.
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The Explicit Formula and Parity for Some Generalized Euler Functions

2021-01-01
Shichun Yang , Qunying Liao , Shan Du +1 more
Based on elementary methods and techniques, the explicit formula for the generalized Euler function $\varphi_{e}(n)(e=8,12)$ is given, and then a sufficient and necessary condition for $\varphi_{8}(n)$ or $\varphi_{12}(n)$ to be … Based on elementary methods and techniques, the explicit formula for the generalized Euler function $\varphi_{e}(n)(e=8,12)$ is given, and then a sufficient and necessary condition for $\varphi_{8}(n)$ or $\varphi_{12}(n)$ to be odd is obtained, respectively.
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The $[1,0]$-twisted generalized Reed-Solomon code

2022-01-01
Canze Zhu , Qunying Liao
In this paper, we not only give the parity check matrix of the $[1,0]$-twisted generalized Reed-Solomon (in short, TGRS) code, but also determine the weight distribution. Especially, we show that … In this paper, we not only give the parity check matrix of the $[1,0]$-twisted generalized Reed-Solomon (in short, TGRS) code, but also determine the weight distribution. Especially, we show that the $[1,0]$-TGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the $[1,0]$-TGRS code to be self-orthogonal, and then construct several classes of self-dual or almost self-dual $[1,0]$-TGRS codes. Finally, basing on these self-dual or almost self-dual $[1,0]$-TGRS codes, we obtain some LCD $[1,0]$-TGRS codes.
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Finding normal bases over finite fields with prescribed trace self-orthogonal relations

2013-01-01
Xiyong Zhang , Rongquan Feng , Qunying Liao +1 more
Normal bases and self-dual normal bases over finite fields have been found to be very useful in many fast arithmetic computations. It is well-known that there exists a self-dual normal … Normal bases and self-dual normal bases over finite fields have been found to be very useful in many fast arithmetic computations. It is well-known that there exists a self-dual normal basis of $\mathbb{F}_{2^n}$ over $\mathbb{F}_2$ if and only if $4\nmid n$. In this paper, we prove there exists a normal element $\alpha$ of $\mathbb{F}_{2^n}$ over $\mathbb{F}_{2}$ corresponding to a prescribed vector $a=(a_0,a_1,...,a_{n-1})\in \mathbb{F}_2^n$ such that $a_i={Tr}_{2^n|2}(\alpha^{1+2^i})$ for $0\leq i\leq n-1$, where $n$ is a 2-power or odd, if and only if the given vector $a$ is symmetric ($a_i=a_{n-i}$ for all $i, 1\leq i\leq n-1$), and one of the following is true. 1) $n=2^s\geq 4$, $a_0=1$, $a_{n/2}=0$, $\sum\limits_{1\leq i\leq n/2-1, (i,2)=1}a_i=1$; 2) $n$ is odd, $(\sum\limits_{0\leq i\leq n-1}a_ix^i,x^n-1)=1$. Furthermore we give an algorithm to obtain normal elements corresponding to prescribed vectors in the above two cases. For a general positive integer $n$ with $4|n$, some necessary conditions for a vector to be the corresponding vector of a normal element of $\mathbb{F}_{2^n}$ over $\mathbb{F}_{2}$ are given. And for all $n$ with $4|n$, we prove that there exists a normal element of $\mathbb{F}_{2^n}$ over $\mathbb{F}_2$ such that the Hamming weight of its corresponding vector is 3, which is the lowest possible Hamming weight.
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On Some Special Normal Bases over Finite Fields

2005-01-01
Qunying Liao
Let q be a power of a prime p and n be a positive integer,K=F_q be the finit field with q elements and F=F_q~n be the n-th extension of K.If … Let q be a power of a prime p and n be a positive integer,K=F_q be the finit field with q elements and F=F_q~n be the n-th extension of K.If N={α_i｜i=0,1,0…,n-1}is a normal basis of F over F_q,B={β_i｜i=0,1,…,n-1}is the dual basis of N.Then the two sufficcient adn neccesary conditions for a,b∈F_q such that β=a+bα are given in this note.The author also gets the relationship between the mnltiplication tables of N and B.

On the r-th Power Additive Complement Function of the Positive Integer n

2013-01-01
Qunying Liao
Let n be a positive integer,and br(n) be the r-th power additive complement function of n.By the Abel's identity,the explicit asymptotic formulas on the mean value for the hybrid functions … Let n be a positive integer,and br(n) be the r-th power additive complement function of n.By the Abel's identity,the explicit asymptotic formulas on the mean value for the hybrid functions nkφl(n+br(n)),φl(n)(n+br(n)) and φl(n)br(n)are obtained,which improve the existing results on the r-th power additive complement.

On t-Euler Pretty Number Pairs

2007-01-01
Qunying Liao
Let n,a,b,t be positive integers,and φ(n) be the Euler function of n.A pair of numbers(t,a,b) is called a pair of t-Euler pretty numbers,if φ(a)=b/t and φ(b)=a/t.In this paper,by using an … Let n,a,b,t be positive integers,and φ(n) be the Euler function of n.A pair of numbers(t,a,b) is called a pair of t-Euler pretty numbers,if φ(a)=b/t and φ(b)=a/t.In this paper,by using an elementary and brief method we prove the existence of t-Euler pretty number pairs and determined that all t-Euler pretty number pairs are only(1,1,1),(2,2α,2α),(3,2α·3β,2α·3β), where α,β are positive integers.

A Class of Special Normal Bases and Their Dual Bases Over Finite Fields With Even Characteristic

2015-05-25
Bo Li , Qunying Liao

On Constacyclic Codes over $$\boldsymbol{Z}_{\boldsymbol{p}_{1}\boldsymbol{p}_{2}\cdots\boldsymbol{p}_{\boldsymbol{t}}}$$

2019-06-14
Derong Xie , Qunying Liao

New Deep Holes of Generalized Reed-Solomon Codes

2012-05-30
Jun Zhang , Fang‐Wei Fu , Qunying Liao
Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong \cite{WH} found a new class of deep holes for standard Reed-Solomon codes. In … Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong \cite{WH} found a new class of deep holes for standard Reed-Solomon codes. In the present paper, we give a concise method to obtain a new class of deep holes for generalized Reed-Solomon codes. In particular, for standard Reed-Solomon codes, we get the new class of deep holes given in \cite{WH}. Li and Wan \cite{L.W1} studied deep holes of generalized Reed-Solomon codes $GRS_{k}(\f,D)$ and characterized deep holes defined by polynomials of degree $k+1$. They showed that this problem is reduced to be a subset sum problem in finite fields. Using the method of Li and Wan, we obtain some new deep holes for special Reed-Solomon codes over finite fields with even characteristic. Furthermore, we study deep holes of the extended Reed-Solomon code, i.e., $D=\f$ and show polynomials of degree $k+2$ can not define deep holes.

Complete weight enumerators for several classes of two-weight and three-weight linear codes

2021-01-01
Canze Zhu , Qunying Liao
In this paper, for an odd prime $p$, by extending Li et al.'s construction \cite{CL2016}, several classes of two-weight and three-weight linear codes over the finite field $\mathbb{F}_p$ are constructed … In this paper, for an odd prime $p$, by extending Li et al.'s construction \cite{CL2016}, several classes of two-weight and three-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from a defining set, and then their complete weight enumerators are determined by using Weil sums. Furthermore, we show that some examples of these codes are optimal or almost optimal with respect to the Griesmer bound. Our results generalize the corresponding results in \cite{CL2016, GJ2019}.

Two Constructions for Minimal Ternary Linear Codes.

2021-07-11
Haibo Liu , Qunying Liao
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and … Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and determining their weight distributions have been an interesting research topic in coding theory and cryptography. In this paper, basing on exponential sums and Krawtchouk polynomials, we first prove that $g_{(m,k)}$ in \cite{Heng-Ding-Zhou}, which is the characteristic function of some subset in $\mathbb{F}_3^m$, can be generalized to be $f{(m,k)}$ for obtaining a minimal linear code violating the Ashikhmin-Barg condition; secondly, we employ $\overline{g}_{(m,k)}$ to construct a class of ternary minimal linear codes violating the Ashikhmin-Barg condition, whose minimal distance is better than that of codes in \cite{Heng-Ding-Zhou}.

Two new classes of projective two-weight linear codes

2021-01-01
Canze Zhu , Qunying Liao
In this paper, for an odd prime $p$, several classes of two-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from the defining sets, and then their complete weight … In this paper, for an odd prime $p$, several classes of two-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from the defining sets, and then their complete weight distributions are determined by employing character sums. These codes can be suitable for applications in secret sharing schemes. Furthermore, two new classes of projective two-weight codes are obtained, and then two new classes of strongly regular graphs are given.

The $b$-weight distribution for MDS codes

2021-01-01
Canze Zhu , Qunying Liao
For a positive integer $b\ge2$, the $b$-symbol code is a new coding framework proposed to combat $b$-errors in $b$-symbol read channels. Especially, the $2$-symbol code is called a symbol-pair code. … For a positive integer $b\ge2$, the $b$-symbol code is a new coding framework proposed to combat $b$-errors in $b$-symbol read channels. Especially, the $2$-symbol code is called a symbol-pair code. Remarkably, a classical maximum distance separable (MDS) code is also an MDS $b$-symbol code. Recently, for any MDS code $\mathcal{C}$, Ma and Luo determined the symbol-pair weight distribution of $\mathcal{C}$. In this paper, by calculating the number of solutions for some equations and utilizing some shortened codes of $\mathcal{C}$, we give the connection between the $b$-weight distribution and the number of codewords in shortened codes of $\mathcal{C}$ with special shape. Furthermore, note that shortened codes of $\mathcal{C}$ are also MDS codes, the number of these codewords with special shape are also determined by the shorten method. From the above calculation, the $b$-weight distribution of $\mathcal{C}$ is determined. Our result generalies the corresonding result of Ma and Luo.

A new class of MDS symbol-pair codes

2021-01-01
Canze Zhu , Qunying Liao
The symbol-pair code is a new coding framework proposed to guard against pair-errors in symbol-pair read channels. Especially, a symbol-pair code with the parameters achieving the Singleton-type bound is called … The symbol-pair code is a new coding framework proposed to guard against pair-errors in symbol-pair read channels. Especially, a symbol-pair code with the parameters achieving the Singleton-type bound is called an MDS symbol-pair code. In this paper, inspiring by the classical construction for Reed-Solomon codes, for any $3\le k<m\le q-2$ and $m_1=\Big\lfloor{\tiny\frac{m}{\lfloor\frac{k-1}{2}\rfloor}}\Big\rfloor$, we construct a class of $q$-ary MDS symbol-pair codes with dimension $k$ and length $n$ $(n=m+m_1, m+m_1-1)$, where $q$ is a prime power. Furthermore, for $k\in\{3,4\}$, the symbol-pair weight distributions for these codes are determined by enumerating the number of polynomials with given roots.

The construction for several few-weight linear codes and their applications

2021-12-11
Canze Zhu , Qunying Liao
In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t=3,5,7)$ are obtained based on some defining sets, and then … In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t=3,5,7)$ are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially, for $m = 3$, a class of MDS codes with parameters $[p,3,p-2]$ are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and $s$-sum sets for any odd $s>1$.
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Self-orthogonal generalized twisted Reed-Solomon codes

2022-01-01
Canze Zhu , Qunying Liao
In this paper, by calculating the dual code of the Schur square for the standard twisted Reed-Solomon code, we give a sufficient and necessary condition for the generalized twisted Reed-Solomon … In this paper, by calculating the dual code of the Schur square for the standard twisted Reed-Solomon code, we give a sufficient and necessary condition for the generalized twisted Reed-Solomon code with $h+t\le k-1$ to be self-orthogonal, where $k$ is dimension, $h$ is hook and $t$ is twist. And then, we show that there is no self-orthogonal generalized twisted Reed-Solomon code under some conditions. Furthermore, several classes of self-orthogonal generalized twisted Reed-Solomon codes are constructed, and some of these codes are non-GRS self-orthogonal MDS codes or NMDS codes.

A New Constructions of Minimal Binary Linear Codes

2022-01-01
Haibo Liu , Qunying Liao
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition … Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for binary linear codes with dimension $m+2$ is presented, then a necessary and sufficient condition for this binary linear code to be minimal is derived. Based on this condition and exponential sums, a new class of minimal binary linear codes violating the Ashikhmin-Barg condition is obtained, and then their weight enumerators are determined.

The complete weight enumerator of the Reed-Solomon code with dimension two or three

2021-01-01
Canze Zhu , Qunying Liao
It is well-known that Reed-Solomon codes and extended Reed-Solomon codes are two special classes of MDS codes with wide applications in practice. The complete weight enumerators of these codes are … It is well-known that Reed-Solomon codes and extended Reed-Solomon codes are two special classes of MDS codes with wide applications in practice. The complete weight enumerators of these codes are very important for determining the capability of both error-detection and error-correction. In this paper, for any positive integer $m$ and prime $p$, basing on the character sums, we determine the complete weight enumerators of the Reed-Solomon code and the extended Reed-Solomon code with dimension $k$ $(k=2,3)$ over $\mathbb{F}_{p^m}$, explictly, which are generalizations of the corresponding results in \cite{BK91,K04}.
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Constructions For Several Few-weight Linear Codes And Their Applications

2021-01-01
Canze Zhu , Qunying Liao
In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t=3,5,7)$ are obtained based on some defining sets, and then … In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t=3,5,7)$ are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially for $m = 3$, a class of MDS codes with parameters $[p,3,p-2]$ are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and $s$-sum sets for any odd $s>1$.
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The $(+)$-extended twisted generalized Reed-Solomon code

2022-01-01
Canze Zhu , Qunying Liao
In this paper, we give a parity check matrix for the $(+)$-extended twisted generalized Reed Solomon (in short, ETGRS) code, and then not only prove that it is MDS or … In this paper, we give a parity check matrix for the $(+)$-extended twisted generalized Reed Solomon (in short, ETGRS) code, and then not only prove that it is MDS or NMDS, but also determine the weight distribution. Especially, based on Schur method, we show that the $(+)$-ETGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the $(+)$-ETGRS code to be self-orthogonal, and then construct several classes of self-dual $(+)$-TGRS codes and almost self-dual $(+)$-ETGRS codes.
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Several classes of projective few-weight linear codes and their applications

2022-01-01
Canze Zhu , Qunying Liao
It is well-known that few-weight linear codes have better applications in secret sharing schemes \cite{JY2006,CC2005}.In particular, projective two-weight codes are very precious as they are closely related to finite projective … It is well-known that few-weight linear codes have better applications in secret sharing schemes \cite{JY2006,CC2005}.In particular, projective two-weight codes are very precious as they are closely related to finite projective spaces, strongly regular graphs and combinatorial designs \cite{RC1986,CD2018,P1972}. Here, we present the following two applications.
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The $ b $-weight distribution for MDS codes

2023-08-22
Canze Zhu , Qunying Liao
For a positive integer $ b\ge2 $, the $ b $-symbol code is a new coding framework proposed to combat $ b $-errors in $ b $-symbol read channels. Especially, … For a positive integer $ b\ge2 $, the $ b $-symbol code is a new coding framework proposed to combat $ b $-errors in $ b $-symbol read channels. Especially, the $ 2 $-symbol code is called the symbol-pair code. Remarkably, a classical maximum distance separable (MDS) code is also an MDS $ b $-symbol code. Recently, for any MDS code $ \mathcal{C} $, Ma and Luo determined the symbol-pair weight distribution of $ \mathcal{C} $. In this paper, by calculating the number of solutions for some equations and utilizing some shortened codes of $ \mathcal{C} $, we give the connection between the $ b $-weight distribution and the number of codewords in shortened codes of $ \mathcal{C} $ with special shape. Note that shortened codes of $ \mathcal{C} $ are also MDS codes, the number of these codewords with special shape are also determined by the shorten method and the number of $ r $-combinations of the multiset problem. From the above calculation, the $ b $-weight distribution of $ \mathcal{C} $ is determined. Our result generalizes the corresponding result of Ma and Luo [15].
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Infinite families $2$-designs from binary projective three-weight codes

2023-01-01
Canze Zhu , Qunying Liao , Haibo Liu
Combinatorial designs are closely related to linear codes. In recent year, there are a lot of $t$-designs constructed from certain linear codes. In this paper, we aim to construct $2$-designs … Combinatorial designs are closely related to linear codes. In recent year, there are a lot of $t$-designs constructed from certain linear codes. In this paper, we aim to construct $2$-designs from binary three-weight codes. For any binary three-weight code $\mathcal{C}$ with length $n$, let $A_{n}(\mathcal{C})$ be the number of codewords in $\mathcal{C}$ with Hamming weight $n$, then we show that $\mathcal{C}$ holds $2$-designs when $\mathcal{C}$ is projective and $A_{n}(\mathcal{C})=1$. Furthermore, by extending some certain binary projective two-weight codes and basing on the defining set method, we construct two classes of binary projective three-weight codes which are suitable for holding $2$-designs.
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The explicit formula and parity for some generalized Euler functions

2024-01-01
Shichun Yang , Qunying Liao , Shan Du +1 more
&lt;abstract&gt;&lt;p&gt;Utilizing elementary methods and techniques, the explicit formula for the generalized Euler function $ \varphi_{e}(n)(e = 8, 12) $ has been developed. Additionally, a sufficient and necessary condition for $ … &lt;abstract&gt;&lt;p&gt;Utilizing elementary methods and techniques, the explicit formula for the generalized Euler function $ \varphi_{e}(n)(e = 8, 12) $ has been developed. Additionally, a sufficient and necessary condition for $ \varphi_{8}(n) $ or $ \varphi_{12}(n) $ to be odd has been obtained, respectively.&lt;/p&gt;&lt;/abstract&gt;
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Two_Generalizations_for_Quadratic_Residue_Codes_over_Finite_Fields

2020-01-01
Qunying Liao , Yuanbo Liu
It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and … It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and $p,p_{1},\ldots, p_{g}$ be distinct odd primes, the present paper generalizes the constructions for the quadratic residue code with length $p$ to be the length $n=p_{1}\cdots p_{g}$, and to be the case $m$-th residue codes with length $p$ over finite fields, where $m\geq 2$ is a positive integer. Furthermore, a criterion for that these codes are self-orthogonal or complementary dual is obtained, and then the corresponding counting formula are given. In particular, the minimum distance of all 24 quaternary quadratic residue codes $[15,8]$ are determined.

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

2025-05-08
Dongmei Huang , Qunying Liao , Gaohua Tang +1 more

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

2025-05-08
Dongmei Huang , Qunying Liao , Gaohua Tang +1 more

A class of double-twisted generalized Reed-Solomon codes

2024-03-01
Canze Zhu , Qunying Liao

The explicit formula and parity for some generalized Euler functions

2024-01-01
Shichun Yang , Qunying Liao , Shan Du +1 more
&lt;abstract&gt;&lt;p&gt;Utilizing elementary methods and techniques, the explicit formula for the generalized Euler function $ \varphi_{e}(n)(e = 8, 12) $ has been developed. Additionally, a sufficient and necessary condition for $ … &lt;abstract&gt;&lt;p&gt;Utilizing elementary methods and techniques, the explicit formula for the generalized Euler function $ \varphi_{e}(n)(e = 8, 12) $ has been developed. Additionally, a sufficient and necessary condition for $ \varphi_{8}(n) $ or $ \varphi_{12}(n) $ to be odd has been obtained, respectively.&lt;/p&gt;&lt;/abstract&gt;
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The (+)-extended twisted generalized Reed-Solomon code

2023-10-17
Canze Zhu , Qunying Liao
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The $ b $-weight distribution for MDS codes

2023-08-22
Canze Zhu , Qunying Liao
For a positive integer $ b\ge2 $, the $ b $-symbol code is a new coding framework proposed to combat $ b $-errors in $ b $-symbol read channels. Especially, … For a positive integer $ b\ge2 $, the $ b $-symbol code is a new coding framework proposed to combat $ b $-errors in $ b $-symbol read channels. Especially, the $ 2 $-symbol code is called the symbol-pair code. Remarkably, a classical maximum distance separable (MDS) code is also an MDS $ b $-symbol code. Recently, for any MDS code $ \mathcal{C} $, Ma and Luo determined the symbol-pair weight distribution of $ \mathcal{C} $. In this paper, by calculating the number of solutions for some equations and utilizing some shortened codes of $ \mathcal{C} $, we give the connection between the $ b $-weight distribution and the number of codewords in shortened codes of $ \mathcal{C} $ with special shape. Note that shortened codes of $ \mathcal{C} $ are also MDS codes, the number of these codewords with special shape are also determined by the shorten method and the number of $ r $-combinations of the multiset problem. From the above calculation, the $ b $-weight distribution of $ \mathcal{C} $ is determined. Our result generalizes the corresponding result of Ma and Luo [15].
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Two new classes of projective two-weight linear codes

2023-03-04
Canze Zhu , Qunying Liao
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Infinite families $2$-designs from binary projective three-weight codes

2023-01-01
Canze Zhu , Qunying Liao , Haibo Liu
Combinatorial designs are closely related to linear codes. In recent year, there are a lot of $t$-designs constructed from certain linear codes. In this paper, we aim to construct $2$-designs … Combinatorial designs are closely related to linear codes. In recent year, there are a lot of $t$-designs constructed from certain linear codes. In this paper, we aim to construct $2$-designs from binary three-weight codes. For any binary three-weight code $\mathcal{C}$ with length $n$, let $A_{n}(\mathcal{C})$ be the number of codewords in $\mathcal{C}$ with Hamming weight $n$, then we show that $\mathcal{C}$ holds $2$-designs when $\mathcal{C}$ is projective and $A_{n}(\mathcal{C})=1$. Furthermore, by extending some certain binary projective two-weight codes and basing on the defining set method, we construct two classes of binary projective three-weight codes which are suitable for holding $2$-designs.
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Self-orthogonal generalized twisted Reed-Solomon codes

2022-01-01
Canze Zhu , Qunying Liao
In this paper, by calculating the dual code of the Schur square for the standard twisted Reed-Solomon code, we give a sufficient and necessary condition for the generalized twisted Reed-Solomon … In this paper, by calculating the dual code of the Schur square for the standard twisted Reed-Solomon code, we give a sufficient and necessary condition for the generalized twisted Reed-Solomon code with $h+t\le k-1$ to be self-orthogonal, where $k$ is dimension, $h$ is hook and $t$ is twist. And then, we show that there is no self-orthogonal generalized twisted Reed-Solomon code under some conditions. Furthermore, several classes of self-orthogonal generalized twisted Reed-Solomon codes are constructed, and some of these codes are non-GRS self-orthogonal MDS codes or NMDS codes.

The non-GRS properties for the twisted generalized Reed-Solomon code and its extended code

2022-01-01
Canze Zhu , Qunying Liao
In 2017, Beelen et al. firstly introduced twisted generalized Reed-Solomon (in short, TGRS) codes, and constructed a large subclass of MDS TGRS codes. Later, they proved that TGRS code is … In 2017, Beelen et al. firstly introduced twisted generalized Reed-Solomon (in short, TGRS) codes, and constructed a large subclass of MDS TGRS codes. Later, they proved that TGRS code is non-GRS when the code rate is less than one half. In this letter, basing on the dual code of the TGRS code or the extended TGRS code, by using the Schur product, we prove that almost all of TGRS codes and extended TGRS codes are non-GRS when the code rate more than one half.
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A New Constructions of Minimal Binary Linear Codes

2022-01-01
Haibo Liu , Qunying Liao
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition … Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for binary linear codes with dimension $m+2$ is presented, then a necessary and sufficient condition for this binary linear code to be minimal is derived. Based on this condition and exponential sums, a new class of minimal binary linear codes violating the Ashikhmin-Barg condition is obtained, and then their weight enumerators are determined.

The $(+)$-extended twisted generalized Reed-Solomon code

2022-01-01
Canze Zhu , Qunying Liao
In this paper, we give a parity check matrix for the $(+)$-extended twisted generalized Reed Solomon (in short, ETGRS) code, and then not only prove that it is MDS or … In this paper, we give a parity check matrix for the $(+)$-extended twisted generalized Reed Solomon (in short, ETGRS) code, and then not only prove that it is MDS or NMDS, but also determine the weight distribution. Especially, based on Schur method, we show that the $(+)$-ETGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the $(+)$-ETGRS code to be self-orthogonal, and then construct several classes of self-dual $(+)$-TGRS codes and almost self-dual $(+)$-ETGRS codes.
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The $[1,0]$-twisted generalized Reed-Solomon code

2022-01-01
Canze Zhu , Qunying Liao
In this paper, we not only give the parity check matrix of the $[1,0]$-twisted generalized Reed-Solomon (in short, TGRS) code, but also determine the weight distribution. Especially, we show that … In this paper, we not only give the parity check matrix of the $[1,0]$-twisted generalized Reed-Solomon (in short, TGRS) code, but also determine the weight distribution. Especially, we show that the $[1,0]$-TGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the $[1,0]$-TGRS code to be self-orthogonal, and then construct several classes of self-dual or almost self-dual $[1,0]$-TGRS codes. Finally, basing on these self-dual or almost self-dual $[1,0]$-TGRS codes, we obtain some LCD $[1,0]$-TGRS codes.
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Several classes of projective few-weight linear codes and their applications

2022-01-01
Canze Zhu , Qunying Liao
It is well-known that few-weight linear codes have better applications in secret sharing schemes \cite{JY2006,CC2005}.In particular, projective two-weight codes are very precious as they are closely related to finite projective … It is well-known that few-weight linear codes have better applications in secret sharing schemes \cite{JY2006,CC2005}.In particular, projective two-weight codes are very precious as they are closely related to finite projective spaces, strongly regular graphs and combinatorial designs \cite{RC1986,CD2018,P1972}. Here, we present the following two applications.
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The construction for several few-weight linear codes and their applications

2021-12-11
Canze Zhu , Qunying Liao
In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t=3,5,7)$ are obtained based on some defining sets, and then … In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t=3,5,7)$ are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially, for $m = 3$, a class of MDS codes with parameters $[p,3,p-2]$ are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and $s$-sum sets for any odd $s>1$.
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Complete weight enumerators for several classes of two-weight and three-weight linear codes

2021-07-29
Canze Zhu , Qunying Liao
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Two Constructions for Minimal Ternary Linear Codes.

2021-07-11
Haibo Liu , Qunying Liao
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and … Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and determining their weight distributions have been an interesting research topic in coding theory and cryptography. In this paper, basing on exponential sums and Krawtchouk polynomials, we first prove that $g_{(m,k)}$ in \cite{Heng-Ding-Zhou}, which is the characteristic function of some subset in $\mathbb{F}_3^m$, can be generalized to be $f{(m,k)}$ for obtaining a minimal linear code violating the Ashikhmin-Barg condition; secondly, we employ $\overline{g}_{(m,k)}$ to construct a class of ternary minimal linear codes violating the Ashikhmin-Barg condition, whose minimal distance is better than that of codes in \cite{Heng-Ding-Zhou}.

Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

2021-04-01
Chunming Tang , Yan Qiu , Qunying Liao +1 more
In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, recently introduced by Bonini and Borello, and then completely characterize minimal linear … In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension 3 and t-fold blocking sets with t ≥ 2 in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg's condition are obtained. In addition to this, open problems on the parameters and the weight distributions of some generated linear codes are presented.
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A recursion formula for the generalized Euler function $ φ_e(n) $

2021-01-01
Canze Zhu , Qunying Liao
In this paper, basing on the linear algebra methods and elementary techniques, for any positive integers $ e $ and $ n $, we obtain a recursion formula for the … In this paper, basing on the linear algebra methods and elementary techniques, for any positive integers $ e $ and $ n $, we obtain a recursion formula for the generalized Euler function $ \varphi_e(n) $, which is determined by some matrices related to a congruence equation modulo $ e $. Furthermore, through the recursion formula, we get the explicit formula for $ \varphi_5(n) $. Our results generalize the corresponding results in \cite{A4,A8,A10,A11}.
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Complete weight enumerators for several classes of two-weight and three-weight linear codes

2021-01-01
Canze Zhu , Qunying Liao
In this paper, for an odd prime $p$, by extending Li et al.'s construction \cite{CL2016}, several classes of two-weight and three-weight linear codes over the finite field $\mathbb{F}_p$ are constructed … In this paper, for an odd prime $p$, by extending Li et al.'s construction \cite{CL2016}, several classes of two-weight and three-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from a defining set, and then their complete weight enumerators are determined by using Weil sums. Furthermore, we show that some examples of these codes are optimal or almost optimal with respect to the Griesmer bound. Our results generalize the corresponding results in \cite{CL2016, GJ2019}.

Two new classes of projective two-weight linear codes

2021-01-01
Canze Zhu , Qunying Liao
In this paper, for an odd prime $p$, several classes of two-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from the defining sets, and then their complete weight … In this paper, for an odd prime $p$, several classes of two-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from the defining sets, and then their complete weight distributions are determined by employing character sums. These codes can be suitable for applications in secret sharing schemes. Furthermore, two new classes of projective two-weight codes are obtained, and then two new classes of strongly regular graphs are given.

The $b$-weight distribution for MDS codes

2021-01-01
Canze Zhu , Qunying Liao
For a positive integer $b\ge2$, the $b$-symbol code is a new coding framework proposed to combat $b$-errors in $b$-symbol read channels. Especially, the $2$-symbol code is called a symbol-pair code. … For a positive integer $b\ge2$, the $b$-symbol code is a new coding framework proposed to combat $b$-errors in $b$-symbol read channels. Especially, the $2$-symbol code is called a symbol-pair code. Remarkably, a classical maximum distance separable (MDS) code is also an MDS $b$-symbol code. Recently, for any MDS code $\mathcal{C}$, Ma and Luo determined the symbol-pair weight distribution of $\mathcal{C}$. In this paper, by calculating the number of solutions for some equations and utilizing some shortened codes of $\mathcal{C}$, we give the connection between the $b$-weight distribution and the number of codewords in shortened codes of $\mathcal{C}$ with special shape. Furthermore, note that shortened codes of $\mathcal{C}$ are also MDS codes, the number of these codewords with special shape are also determined by the shorten method. From the above calculation, the $b$-weight distribution of $\mathcal{C}$ is determined. Our result generalies the corresonding result of Ma and Luo.

Self-dual twisted generalized Reed-Solomon codes

2021-01-01
Canze Zhu , Qunying Liao
In this paper, by using some properties for linear algebra methods, the parity-check matrices for twisted generalized Reed-Solomon codes with any given hook $h$ and twist $t$ are presented, and … In this paper, by using some properties for linear algebra methods, the parity-check matrices for twisted generalized Reed-Solomon codes with any given hook $h$ and twist $t$ are presented, and then a sufficient and necessary condition for the twisted generalized Reed-Solomon code with $h\ge t$ to be self-dual is given. Furthermore, several classes of self-dual codes with small Singleton defect are constructed based on twisted generalized Reed-Solomon codes, especially some of these self-dual codes are MDS or NMDS.
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A new class of MDS symbol-pair codes

2021-01-01
Canze Zhu , Qunying Liao
The symbol-pair code is a new coding framework proposed to guard against pair-errors in symbol-pair read channels. Especially, a symbol-pair code with the parameters achieving the Singleton-type bound is called … The symbol-pair code is a new coding framework proposed to guard against pair-errors in symbol-pair read channels. Especially, a symbol-pair code with the parameters achieving the Singleton-type bound is called an MDS symbol-pair code. In this paper, inspiring by the classical construction for Reed-Solomon codes, for any $3\le k<m\le q-2$ and $m_1=\Big\lfloor{\tiny\frac{m}{\lfloor\frac{k-1}{2}\rfloor}}\Big\rfloor$, we construct a class of $q$-ary MDS symbol-pair codes with dimension $k$ and length $n$ $(n=m+m_1, m+m_1-1)$, where $q$ is a prime power. Furthermore, for $k\in\{3,4\}$, the symbol-pair weight distributions for these codes are determined by enumerating the number of polynomials with given roots.

The Explicit Formula and Parity for Some Generalized Euler Functions

2021-01-01
Shichun Yang , Qunying Liao , Shan Du +1 more
Based on elementary methods and techniques, the explicit formula for the generalized Euler function $\varphi_{e}(n)(e=8,12)$ is given, and then a sufficient and necessary condition for $\varphi_{8}(n)$ or $\varphi_{12}(n)$ to be … Based on elementary methods and techniques, the explicit formula for the generalized Euler function $\varphi_{e}(n)(e=8,12)$ is given, and then a sufficient and necessary condition for $\varphi_{8}(n)$ or $\varphi_{12}(n)$ to be odd is obtained, respectively.
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The complete weight enumerator of the Reed-Solomon code with dimension two or three

2021-01-01
Canze Zhu , Qunying Liao
It is well-known that Reed-Solomon codes and extended Reed-Solomon codes are two special classes of MDS codes with wide applications in practice. The complete weight enumerators of these codes are … It is well-known that Reed-Solomon codes and extended Reed-Solomon codes are two special classes of MDS codes with wide applications in practice. The complete weight enumerators of these codes are very important for determining the capability of both error-detection and error-correction. In this paper, for any positive integer $m$ and prime $p$, basing on the character sums, we determine the complete weight enumerators of the Reed-Solomon code and the extended Reed-Solomon code with dimension $k$ $(k=2,3)$ over $\mathbb{F}_{p^m}$, explictly, which are generalizations of the corresponding results in \cite{BK91,K04}.
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Constructions For Several Few-weight Linear Codes And Their Applications

2021-01-01
Canze Zhu , Qunying Liao
In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t=3,5,7)$ are obtained based on some defining sets, and then … In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t=3,5,7)$ are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially for $m = 3$, a class of MDS codes with parameters $[p,3,p-2]$ are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and $s$-sum sets for any odd $s>1$.
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Two_Generalizations_for_Quadratic_Residue_Codes_over_Finite_Fields

2020-01-01
Qunying Liao , Yuanbo Liu
It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and … It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and $p,p_{1},\ldots, p_{g}$ be distinct odd primes, the present paper generalizes the constructions for the quadratic residue code with length $p$ to be the length $n=p_{1}\cdots p_{g}$, and to be the case $m$-th residue codes with length $p$ over finite fields, where $m\geq 2$ is a positive integer. Furthermore, a criterion for that these codes are self-orthogonal or complementary dual is obtained, and then the corresponding counting formula are given. In particular, the minimum distance of all 24 quaternary quadratic residue codes $[15,8]$ are determined.

On Constacyclic Codes over $$\boldsymbol{Z}_{\boldsymbol{p}_{1}\boldsymbol{p}_{2}\cdots\boldsymbol{p}_{\boldsymbol{t}}}$$

2019-06-14
Derong Xie , Qunying Liao

Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

2019-01-01
Chunming Tang , Yan Qiu , Qunying Liao +1 more
In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize … In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension $3$ and $t$-fold blocking sets with $t\ge 2$ in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. This confirms a recent conjecture by Alfarano, Borello and Neri in [a geometric characterization of minimal codes and their asymptotic performance, arXiv:1911.11738, 2019] about a lower bound of the minimum distance of a minimal code. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg's condition are obtained. In addition to this, the weight distributions of two subfamilies of the proposed minimal linear codes are established. Open problems are also presented.
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A CLASS OF NEW NEAR-PERFECT NUMBERS

2015-06-01
Yanbin Li , Qunying Liao
Let <TEX>${\alpha}$</TEX> be a positive integer, and let <TEX>$p_1$</TEX>, <TEX>$p_2$</TEX> be two distinct prime numbers with <TEX>$p_1$</TEX> < <TEX>$p_2$</TEX>. By using elementary methods, we give two equivalent conditions of all … Let <TEX>${\alpha}$</TEX> be a positive integer, and let <TEX>$p_1$</TEX>, <TEX>$p_2$</TEX> be two distinct prime numbers with <TEX>$p_1$</TEX> < <TEX>$p_2$</TEX>. By using elementary methods, we give two equivalent conditions of all even near-perfect numbers in the form <TEX>$2^{\alpha}p_1p_2$</TEX> and <TEX>$2^{\alpha}p_1^2p_2$</TEX>, and obtain a lot of new near-perfect numbers which involve some special kinds of prime number pairs. One kind is exactly the new Mersenne conjecture's prime number pair. Another kind has the form <TEX>$p_1=2^{{\alpha}+1}-1$</TEX> and <TEX>$p_2={\frac{p^2_1+p_1+1}{3}}$</TEX>, where the former is a Mersenne prime and the latter's behavior is very much like a Fermat number.
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A Class of Special Normal Bases and Their Dual Bases Over Finite Fields With Even Characteristic

2015-05-25
Bo Li , Qunying Liao

Finding normal bases over finite fields with prescribed trace self-orthogonal relations

2014-02-17
Xiyong Zhang , Rongquan Feng , Qunying Liao +1 more
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Finding normal bases over finite fields with prescribed trace self-orthogonal relations

2013-01-01
Xiyong Zhang , Rongquan Feng , Qunying Liao +1 more
Normal bases and self-dual normal bases over finite fields have been found to be very useful in many fast arithmetic computations. It is well-known that there exists a self-dual normal … Normal bases and self-dual normal bases over finite fields have been found to be very useful in many fast arithmetic computations. It is well-known that there exists a self-dual normal basis of $\mathbb{F}_{2^n}$ over $\mathbb{F}_2$ if and only if $4\nmid n$. In this paper, we prove there exists a normal element $\alpha$ of $\mathbb{F}_{2^n}$ over $\mathbb{F}_{2}$ corresponding to a prescribed vector $a=(a_0,a_1,...,a_{n-1})\in \mathbb{F}_2^n$ such that $a_i={Tr}_{2^n|2}(\alpha^{1+2^i})$ for $0\leq i\leq n-1$, where $n$ is a 2-power or odd, if and only if the given vector $a$ is symmetric ($a_i=a_{n-i}$ for all $i, 1\leq i\leq n-1$), and one of the following is true. 1) $n=2^s\geq 4$, $a_0=1$, $a_{n/2}=0$, $\sum\limits_{1\leq i\leq n/2-1, (i,2)=1}a_i=1$; 2) $n$ is odd, $(\sum\limits_{0\leq i\leq n-1}a_ix^i,x^n-1)=1$. Furthermore we give an algorithm to obtain normal elements corresponding to prescribed vectors in the above two cases. For a general positive integer $n$ with $4|n$, some necessary conditions for a vector to be the corresponding vector of a normal element of $\mathbb{F}_{2^n}$ over $\mathbb{F}_{2}$ are given. And for all $n$ with $4|n$, we prove that there exists a normal element of $\mathbb{F}_{2^n}$ over $\mathbb{F}_2$ such that the Hamming weight of its corresponding vector is 3, which is the lowest possible Hamming weight.
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On the r-th Power Additive Complement Function of the Positive Integer n

2013-01-01
Qunying Liao
Let n be a positive integer,and br(n) be the r-th power additive complement function of n.By the Abel's identity,the explicit asymptotic formulas on the mean value for the hybrid functions … Let n be a positive integer,and br(n) be the r-th power additive complement function of n.By the Abel's identity,the explicit asymptotic formulas on the mean value for the hybrid functions nkφl(n+br(n)),φl(n)(n+br(n)) and φl(n)br(n)are obtained,which improve the existing results on the r-th power additive complement.

New Deep Holes of Generalized Reed-Solomon Codes

2012-05-30
Jun Zhang , Fang‐Wei Fu , Qunying Liao
Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong \cite{WH} found a new class of deep holes for standard Reed-Solomon codes. In … Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong \cite{WH} found a new class of deep holes for standard Reed-Solomon codes. In the present paper, we give a concise method to obtain a new class of deep holes for generalized Reed-Solomon codes. In particular, for standard Reed-Solomon codes, we get the new class of deep holes given in \cite{WH}. Li and Wan \cite{L.W1} studied deep holes of generalized Reed-Solomon codes $GRS_{k}(\f,D)$ and characterized deep holes defined by polynomials of degree $k+1$. They showed that this problem is reduced to be a subset sum problem in finite fields. Using the method of Li and Wan, we obtain some new deep holes for special Reed-Solomon codes over finite fields with even characteristic. Furthermore, we study deep holes of the extended Reed-Solomon code, i.e., $D=\f$ and show polynomials of degree $k+2$ can not define deep holes.

New Deep Holes of Generalized Reed-Solomon Codes

2012-01-01
Jun Zhang , Fang‐Wei Fu , Qunying Liao
Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong \cite{WH} found a new class of deep holes for standard Reed-Solomon codes. In … Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong \cite{WH} found a new class of deep holes for standard Reed-Solomon codes. In the present paper, we give a concise method to obtain a new class of deep holes for generalized Reed-Solomon codes. In particular, for standard Reed-Solomon codes, we get the new class of deep holes given in \cite{WH}. Li and Wan \cite{L.W1} studied deep holes of generalized Reed-Solomon codes $GRS_{k}(\f,D)$ and characterized deep holes defined by polynomials of degree $k+1$. They showed that this problem is reduced to be a subset sum problem in finite fields. Using the method of Li and Wan, we obtain some new deep holes for special Reed-Solomon codes over finite fields with even characteristic. Furthermore, we study deep holes of the extended Reed-Solomon code, i.e., $D=\f$ and show polynomials of degree $k+2$ can not define deep holes.
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On the Distribution of Normal Bases Over Finite Fields

2010-01-01
Qunying Liao

On an identity associated with Weil's estimate and its applications

2009-02-22
Yuan He , Qunying Liao

Some Congruences Involving Euler Numbers

2008-08-01
Yuan He , Qunying Liao
In this paper, we obtain some explicit congruences for Euler numbers modulo an odd prime power in an elementary way. The classical Bernoulli polynomials Bn(x) and Euler polynomials En(x) are … In this paper, we obtain some explicit congruences for Euler numbers modulo an odd prime power in an elementary way. The classical Bernoulli polynomials Bn(x) and Euler polynomials En(x) are usually deflned by the exponential generating functions: te xt e t i 1 = 1 X n=0 Bn(x) t n n! and 2e xt e t + 1 = 1 X n=0 En(x) t n n! :

On the divisibility of power LCM matrices by power GCD matrices

2007-03-01
Jianrong Zhao , Shaofang Hong , Qunying Liao +1 more
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On t-Euler Pretty Number Pairs

2007-01-01
Qunying Liao
Let n,a,b,t be positive integers,and φ(n) be the Euler function of n.A pair of numbers(t,a,b) is called a pair of t-Euler pretty numbers,if φ(a)=b/t and φ(b)=a/t.In this paper,by using an … Let n,a,b,t be positive integers,and φ(n) be the Euler function of n.A pair of numbers(t,a,b) is called a pair of t-Euler pretty numbers,if φ(a)=b/t and φ(b)=a/t.In this paper,by using an elementary and brief method we prove the existence of t-Euler pretty number pairs and determined that all t-Euler pretty number pairs are only(1,1,1),(2,2α,2α),(3,2α·3β,2α·3β), where α,β are positive integers.

On Some Special Normal Bases over Finite Fields

2005-01-01
Qunying Liao
Let q be a power of a prime p and n be a positive integer,K=F_q be the finit field with q elements and F=F_q~n be the n-th extension of K.If … Let q be a power of a prime p and n be a positive integer,K=F_q be the finit field with q elements and F=F_q~n be the n-th extension of K.If N={α_i｜i=0,1,0…,n-1}is a normal basis of F over F_q,B={β_i｜i=0,1,…,n-1}is the dual basis of N.Then the two sufficcient adn neccesary conditions for a,b∈F_q such that β=a+bα are given in this note.The author also gets the relationship between the mnltiplication tables of N and B.
Coauthor Papers Together
Canze Zhu 20
Haibo Liu 3
Jun Zhang 2
Yan Qiu 2
Chunming Tang 2
Rongquan Feng 2
Fang‐Wei Fu 2
Xuhong Gao 2
Xiyong Zhang 2
Zhengchun Zhou 2
Hui-li Wang 2
Shan Du 2
Shichun Yang 2
Yuan He 1
Bo Li 1
Shaofang Hong 1
Yuan He 1
Yuanbo Liu 1
Shixin Zhu 1
Yanbin Li 1
Jianrong Zhao 1
Derong Xie 1
Dongmei Huang 1
K. P. Shum 1
Gaohua Tang 1

Linear Codes With Two or Three Weights From Weakly Regular Bent Functions

2016-01-18
Chunming Tang , Nian Li , Yanfeng Qi +2 more
Linear codes with a few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the … Linear codes with a few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of constructing two-weight and three-weight linear codes of Ding et al. and Zhou et al. to general weakly regular bent functions and determines the weight distributions of these linear codes. It solves an open problem proposed by Ding et al. Furthermore, this paper constructs new linear codes with two or three weights and presents their weight distributions. They contain some optimal codes meeting certain bound on linear codes.
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A Class of Two-Weight and Three-Weight Codes and Their Applications in Secret Sharing

2015-08-27
Kelan Ding , Cunsheng Ding
In this paper, a class of two-weight and three-weight linear codes over GF(p) is constructed, and their application in secret sharing is investigated. Some of the linear codes obtained are … In this paper, a class of two-weight and three-weight linear codes over GF(p) is constructed, and their application in secret sharing is investigated. Some of the linear codes obtained are optimal in the sense that they meet certain bounds on linear codes. These codes have applications also in authentication codes, association schemes, and strongly regular graphs, in addition to their applications in consumer electronics, communication and data storage systems.
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A construction of q-ary linear codes with two weights

2017-08-09
Ziling Heng , Qin Yue
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A Classical Introduction to Modern Number Theory

1982-01-01
Kenneth Ireland , Michael Rosen

Minimal Linear Codes From Characteristic Functions

2020-03-06
Sihem Mesnager , Yanfeng Qi , Hongming Ru +1 more
Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb{F}_q$ to construct minimal linear codes. By … Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb{F}_q$ to construct minimal linear codes. By properties of characteristic functions, we can obtain more minimal binary linear codes from known minimal binary linear codes, which generalizes results of Ding et al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018]. By characteristic functions corresponding to some subspaces of $\mathbb{F}_q$, we obtain many minimal linear codes, which generalizes results of [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018] and [IEEE Trans. Inf. Theory, vol. 65, no. 11, pp. 7067-7078, 2019]. Finally, we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes.
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New MDS self-dual codes over finite fields of odd characteristic

2020-02-26
Xiaolei Fang , Khawla Lebed , Hongwei Liu +1 more
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Minimal linear codes over finite fields

2018-08-23
Ziling Heng , Cunsheng Ding , Zhengchun Zhou
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Constacyclic Symbol-Pair Codes: Lower Bounds and Optimal Constructions

2017-09-18
Bocong Chen , Liren Lin , Hongwei Liu
Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the … Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First, we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai et al. (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second, we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes.
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Maximum distance separable codes for b-symbol read channels

2017-10-27
Baokun Ding , Tao Zhang , Gennian Ge
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On self-dual negacirculant codes of index two and four

2018-01-22
Minjia Shi , Liqin Qian , Patrick Solé
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Construction of MDS Self-dual Codes over Finite Fields

2019-06-18
Khawla Lebed , Hongwei Liu , Jinquan Luo
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New MDS Self-Dual Codes From Generalized Reed—Solomon Codes

2016-12-28
Lingfei Jin , Chaoping Xing
Both Maximum Distance Separable and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, … Both Maximum Distance Separable and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining the existence of q-ary MDS self-dual codes for various lengths has been investigated extensively. The problem is completely solved for the case where q is even. This paper focuses on the case where q is odd. We construct a few classes of new MDS self-dual codes through generalized Reed-Solomon codes. More precisely, we show that for any given even length n, we have a q-ary MDS code as long as q ≡ 1 mod 4 and q is sufficiently large (say q ≥ 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> × n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). Furthermore, we prove that there exists a q-ary MDS self-dual code of length n if q = r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and n satisfies one of the three conditions: 1) n ≤ r and n is even; 2) q is odd and n - 1 is an odd divisor of q - 1; and 3) r ≡ 3 mod 4 and n=2tr for any t ≤ (r - 1)/2.
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Minimal Linear Codes in Odd Characteristic

2019-01-15
Daniele Bartoli , Matteo Bonini
In this paper, we generalize constructions in two recent works of Ding, Heng, and Zhou to any field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , q odd, providing infinite families of minimal … In this paper, we generalize constructions in two recent works of Ding, Heng, and Zhou to any field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , q odd, providing infinite families of minimal codes for which the Ashikhmin-Barg bound does not hold.
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Complete weight enumerators of a family of three-weight linear codes

2016-02-18
Shudi Yang , Zheng‐an Yao
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Maximum Distance Separable Codes for Symbol-Pair Read Channels

2013-10-16
Yeow Meng Chee , Lijun Ji , Han Mao Kiah +2 more
We study (symbol-pair) codes for symbol-pair read channels introduced recently by Cassuto and Blaum (2010). A Singleton-type bound on symbol-pair codes is established and infinite families of optimal symbol-pair codes … We study (symbol-pair) codes for symbol-pair read channels introduced recently by Cassuto and Blaum (2010). A Singleton-type bound on symbol-pair codes is established and infinite families of optimal symbol-pair codes are constructed. These codes are maximum distance separable (MDS) in the sense that they meet the Singleton-type bound. In contrast to classical codes, where all known q-ary MDS codes have length O(q), we show that q-ary MDS symbol-pair codes can have length Ω(q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). In addition, we completely determine the existence of MDS symbol-pair codes for certain parameters.
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Linear codes with two or three weights from quadratic Bent functions

2015-11-04
Zhengchun Zhou , Nian Li , Cuiling Fan +1 more
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Minimal linear codes arising from blocking sets

2020-02-19
Matteo Bonini , Martino Borello
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Two-weight and three-weight linear codes based on Weil sums

2019-02-23
Gaopeng Jian , Zhouchen Lin , Rongquan Feng
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New MDS Euclidean Self-Orthogonal Codes

2020-09-03
Xiaolei Fang , Meiqing Liu , Jinquan Luo
In this paper, two criterions of MDS Euclidean self-orthogonal codes are presented. New MDS Euclidean self-dual codes and self-orthogonal codes are constructed via the criterions. In particular, among our constructions, … In this paper, two criterions of MDS Euclidean self-orthogonal codes are presented. New MDS Euclidean self-dual codes and self-orthogonal codes are constructed via the criterions. In particular, among our constructions, for large square q, about 1/8·q new MDS Euclidean (almost) self-dual codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> can be produced. Moreover, we can construct about 1/4· q new MDS Euclidean self-orthogonal codes with different even lengths n and dimension n/2-1.
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Explicit evaluations of some Weil sums

1998-01-01
Robert S. Coulter
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Multiple Blocking Sets and Arcs in Finite Planes

1996-12-01
Simeon Ball
This paper contains two main results relating to the size of a multiple blocking set in PG(2, q). The first gives a very general lower bound, the second a much … This paper contains two main results relating to the size of a multiple blocking set in PG(2, q). The first gives a very general lower bound, the second a much better lower bound for prime planes. The latter is used to consider maximum sizes of (k, n)-arcs in PG(2, 11) and PG(2, 13), some of which are determined. In addition, a summary is given of the value of mn(2, q) for q ⩽ 13.

Complete b-symbol weight distribution of some irreducible cyclic codes

2022-03-21
Hongwei Zhu , Minjia Shi , Ferruh Özbudak
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An Introduction to the Theory of Numbers. By G. H. Hardy and E. M. Wright. 2nd edition. Pp. xvi, 407 25s. 1945. (Oxford)

1946-02-01
T. A. A. B.

The parameters of minimal linear codes

2021-01-20
Wei Lu , Xia Wu , Xiwang Cao
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Polynomial multiplicities over finite fields and intersection sets

1992-05-01
Aiden A. Bruen

Complete weight enumerators for several classes of two-weight and three-weight linear codes

2021-07-29
Canze Zhu , Qunying Liao
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MDS or NMDS self-dual codes from twisted generalized Reed–Solomon codes

2021-07-12
Daitao Huang , Qin Yue , Yongfeng Niu +1 more
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An inductive construction of minimal codes

2021-02-27
Daniele Bartoli , Matteo Bonini , Burçi̇n Güneş
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Weights of linear codes and strongly regular normed spaces

1972-01-01
P. Delsarte

Blocking Sets in Finite Projective Planes

1971-11-01
Aiden A. Bruen

Further evaluations of Weil sums

1998-01-01
Robert S. Coulter

CONSTRUCTION OF SELF-DUAL INTEGRAL NORMAL BASES IN ABELIAN EXTENSIONS OF FINITE AND LOCAL FIELDS

2010-11-01
Erik Jarl Pickett
Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that Tr … Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that Tr F/E (g(x), h(x)) = δ g, h for g, h ∈ Γ. Bayer-Fluckiger and Lenstra have shown that when char (E) ≠ 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char (E) = 2, then F admits a self-dual normal basis if and only if the exponent of Γ is not divisible by 4. In this paper, we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let K be a finite extension of ℚ p , let L/K be a finite abelian Galois extension of odd degree and let [Formula: see text] be the valuation ring of L. We define A L/K to be the unique fractional [Formula: see text]-ideal with square equal to the inverse different of L/K. It is known that a self-dual integral normal basis exists for A L/K if and only if L/K is weakly ramified. Assuming p ≠ 2, we construct such bases whenever they exist.
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Self-Complementary Normal Bases in Finite Fields

1988-05-01
A. Lempel , M.J. Weinberger
It is shown that $\mathrm{GF} ( q^n )$ has a self complementary normal basis over $\mathrm{GF} ( q )$ if and only if n is odd or $n \equiv 2(\bmod{\text{-}}4)$ … It is shown that $\mathrm{GF} ( q^n )$ has a self complementary normal basis over $\mathrm{GF} ( q )$ if and only if n is odd or $n \equiv 2(\bmod{\text{-}}4)$ and q is even. All existence proofs are constructive and can be readily employed to obtain such bases.

Evaluation of the Hamming weights of a class of linear codes based on Gauss sums

2016-05-21
Ziling Heng , Qin Yue
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Construction of self-dual normal bases and their complexity

2011-11-14
François Arnault , Erik Jarl Pickett , Stéphane Vinatier
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A characterization of flat spaces in a finite geometry and the uniqueness of the hamming and the MacDonald codes

1966-06-01
R. C. Bose , Robert C. Burton

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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A new class of three-weight linear codes from weakly regular plateaued functions.

2017-03-24
Sihem Mesnager , Ferruh Özbudak , Ahmet Sınak
Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in … Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in arbitrary characteristic. To do this, we generalize the recent contribution of Mesnager given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present a new class of binary linear codes with three weights from plateaued Boolean functions and their weight distributions. We next introduce the notion of (weakly) regular plateaued functions in odd characteristic $p$ and give concrete examples of these functions. Moreover, we construct a new class of three-weight linear $p$-ary codes from weakly regular plateaued functions and determine their weight distributions. We finally analyse the constructed linear codes for secret sharing schemes.

A new family of linear maximum rank distance codes

2016-08-01
John Sheekey
In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin … In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes.
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Constructions of maximum distance separable symbol-pair codes using cyclic and constacyclic codes

2016-08-19
Shuxing Li , Gennian Ge
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On Congruences Involving Bernoulli Numbers and the Quotients of Fermat and Wilson

1938-04-01
Emma Lehmer

Lacunary Polynomials, Multiple Blocking Sets and Baer Subplanes

1999-10-01
A. Blokhuis , Leo Storme , Tamás Szőnyi
New lower bounds are given for the size of a point set in a Desarguesian projective plane over a finite field that contains at least a prescribed number s of … New lower bounds are given for the size of a point set in a Desarguesian projective plane over a finite field that contains at least a prescribed number s of points on every line. These bounds are best possible when q is square and s is small compared with q. In this case the smallest set is shown to be the union of disjoint Baer subplanes. The results are based on new results on the structure of certain lacunary polynomials, which can be regarded as a generalization of Rédei's results in the case when the derivative of the polynomial vanishes.

It is indeed a fundamental construction of all linear codes

2016-01-01
Can Xiang
Linear codes are widely employed in communication systems, consumer electronics, and storage devices. All linear codes over finite fields can be generated by a generator matrix. Due to this, the … Linear codes are widely employed in communication systems, consumer electronics, and storage devices. All linear codes over finite fields can be generated by a generator matrix. Due to this, the generator matrix approach is called a fundamental construction of linear codes. This is the only known construction method that can produce all linear codes over finite fields. Recently, a defining-set construction of linear codes over finite fields has attracted a lot of attention, and have been employed to produce a huge number of classes of linear codes over finite fields. It was claimed that this approach can also generate all linear codes over finite fields. But so far, no proof of this claim is given in the literature. The objective of this paper is to prove this claim, and confirm that the defining-set approach is indeed a fundamental approach to constructing all linear codes over finite fields. As a byproduct, a trace representation of all linear codes over finite fields is presented.

Matrices associated with multiplicative functions

1995-02-01
Keith Bourque , Steve Ligh

Matrices Associated with Classes of Arithmetical Functions

1993-11-01
Keith Bourque , Steve Ligh

On perfect and near-perfect numbers

2012-08-13
Paul Pollack , Vladimir Shevelev
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Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes

2009-05-17
Simeon Ball , Szabolcs L. Fancsali

Dual and self-dual negacyclic codes of even length over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math>

2008-09-17
Shixin Zhu , Xiaoshan Kai

Matrices associated with arithmetical functions

1993-05-01
Keith Bourque , Steve Ligh
Let f be an arithmetical function and S={x 1,x 2,…,xn } a set of distinct positive integers. Denote by [f(xi ,xj }] the n×n matrix having f evaluated at the … Let f be an arithmetical function and S={x 1,x 2,…,xn } a set of distinct positive integers. Denote by [f(xi ,xj }] the n×n matrix having f evaluated at the greatest common divisor (xi ,xj ) of xi , and xj as its i j-entry. We will determine conditions on f that will guarantee the matrix [f(xi ,xj )] is positive definite and, in fact, has properties similar to the greatest common divisor (GCD) matrix [(xi ,xj )] where f is the identity function. The set S is gcd-closed if (xi ,xj )∈S for 1≤ i j ≤ n. If S is gcd-closed, we calculate the determinant and (if it is invertible) the inverse of the matrix [f(xi ,xj )]. Among the examples of determinants of this kind are H. J. S. Smith's determinant det[(i,j)].

On the mean value of the two-term exponential sums with Dirichlet characters

2006-08-15
Zhefeng Xu , Tianping Zhang , Wenpeng Zhang