Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case …
Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The $m$-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the $m$-adic residue codes over the quotient ring $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We determine the idempotent generators of the $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We obtain some parameters of optimal $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ with respect to Griesmer bound for rings. Furthermore, we derive a condition for $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ and give some examples to illustrate our findings.
Linear codes with complementary duals (LCD) have a great deal of significance amongst linear codes. Maximum distance separable (MDS) codes are also an important class of linear codes since they …
Linear codes with complementary duals (LCD) have a great deal of significance amongst linear codes. Maximum distance separable (MDS) codes are also an important class of linear codes since they achieve the greatest error correcting and detecting capabilities for fixed length and dimension. The construction of linear codes that are both LCD and MDS is a hard task in coding theory. In this paper, we study the constructions of LCD codes that are MDS from negacyclic codes over finite fields of odd prime power q elements. We construct four families of MDS negacyclic LCD codes of length n|q-1/2, n|q+1/2 and a family of negacyclic LCD codes of length n = q - 1. Furthermore, we obtain five families of q2-ary Hermitian MDS negacyclic LCD codes of length n|(q-1) and four families of Hermitian negacyclic LCD codes of length n = q2 + 1. For both Euclidean and Hermitian cases the dimensions of these codes are determined and for some classes the minimum distances are settled. For the other cases, by studying q and q2-cyclotomic classes we give lower bounds on the minimum distance.
This paper is devoted to the study of the construction of new quantum MDS codes. Based on constacyclic codes over Fq2 , we derive four new families of quantum MDS …
This paper is devoted to the study of the construction of new quantum MDS codes. Based on constacyclic codes over Fq2 , we derive four new families of quantum MDS codes, one of which is an explicit generalization of the construction given in Theorem 7 in [22]. We also extend the result of Theorem 3:3 given in [17].
Linear complementary dual (LCD) codes are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes …
Linear complementary dual (LCD) codes are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings Rk. We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We make a table of lower bounds for this combinatorial function for modest values of the parameters.
<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>
In this paper, we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a noncommutative ring called the skew polynomial …
In this paper, we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a noncommutative ring called the skew polynomial ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ;¿]. After a brief description of the skew polynomial ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ;¿], it is shown that skew QC codes are left submodules of the ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rsl</i> =( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ;¿]/( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xs</i> -1) ) <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> . The notions of generator and parity-check polynomials are given. We also introduce the notion of similar polynomials in the ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ;¿] and show that parity-check polynomials for skew QC codes are unique up to similarity. Our search results lead to the construction of several new codes with Hamming distances exceeding the Hamming distances of the previously best known linear codes with comparable parameters.
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.
For any given positive integer m, a necessary and sufficient condition for the existence of Type-I m-adic constacyclic codes is given. Furthermore, for any given integer s, a necessary and …
For any given positive integer m, a necessary and sufficient condition for the existence of Type-I m-adic constacyclic codes is given. Furthermore, for any given integer s, a necessary and sufficient condition for s to be a multiplier of a Type-I polyadic constacyclic code is given. As an application, some optimal codes from Type-I polyadic constacyclic codes, including generalized Reed-Solomon codes and alternant maximum distance separable codes, are constructed.
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.
We generalize the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. The resulting non commutative rings are no longer left …
We generalize the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. The resulting non commutative rings are no longer left and right Euclidean. Codes that are principal ideals in quotient rings of skew polynomial rings by a two sided ideals are studied. As an application, skew constacyclic self-dual codes over $GR(4, 2)$ are constructed. Euclidean self-dual codes give self-dual $\mathbb Z_4$−codes. Hermitian self-dual codes yield 3−modular lattices and quasi-cyclic self-dual $\mathbb Z_4$−codes.
In this paper, we study skew cyclic codes over the ring R = Fq + uFq + vFq + uvFq, where u 2 = u, v 2 = v, uv …
In this paper, we study skew cyclic codes over the ring R = Fq + uFq + vFq + uvFq, where u 2 = u, v 2 = v, uv = vu, q = p m and p is an odd prime.We investigate the structural properties of skew cyclic codes over R through a decomposition theorem.Furthermore, we give a formula for the number of skew cyclic codes of length n over R.
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.
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.
LCD codes are linear codes that intersect with their dual trivially. Quasi cyclic codes that are LCD are characterized and studied by using their concatenated structure. Some asymptotic results are …
LCD codes are linear codes that intersect with their dual trivially. Quasi cyclic codes that are LCD are characterized and studied by using their concatenated structure. Some asymptotic results are derived. Hermitian LCD codes are introduced to that end and their cyclic subclass is characterized. Constructions of QCCD codes from codes over larger alphabets are given.
Skew polynomial rings over finite fields and over Galois ringshave recently been used to study codes. In this paper, we extend this concept to finite chain rings.Properties of skew constacyclic …
Skew polynomial rings over finite fields and over Galois ringshave recently been used to study codes. In this paper, we extend this concept to finite chain rings.Properties of skew constacyclic codes generated by monic right divisors of $x^n-\lambda$, where $\lambda$ is aunit element, are exhibited. When $\lambda^2=1$, the generators of Euclidean and Hermitian dual codes of suchcodes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Specializing to codes over the ring $\mathbb F$pm$+u\mathbb F$pm, the structure of allskew constacyclic codes is completely determined. This allows us to express the generators ofEuclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of theoriginal codes. An illustration of all skew cyclic codes of length $2$ over $\mathbb F_3 + u\mathbb F_3$ andtheir Euclidean and Hermitian dual codes is also provided.
In 1964, Massey introduced a class of codes with complementary duals which are called Linear Complimentary Dual (LCD for short) codes.He showed that LCD codes have applications in communication system, …
In 1964, Massey introduced a class of codes with complementary duals which are called Linear Complimentary Dual (LCD for short) codes.He showed that LCD codes have applications in communication system, side-channel attack (SCA) and so on.LCD codes have been extensively studied in literature.On the other hand, MDS codes form an optimal family of classical codes which have wide applications in both theory and practice.The main purpose of this paper is to give an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields.We exemplify this construction and obtain several classes of explicit LCD MDS codes for the odd characteristic case.
In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0.$ We give the definition of these codes as subsets …
In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0.$ We give the definition of these codes as subsets of the ring $\mathbb{Z}_{q}^{\alpha}R^{\beta}$. Some structural properties of the skew polynomial ring $ R[x,\Theta]$ are discussed, where $ \Theta$ is an automorphism of $R.$ We describe the generator polynomials of skew constacyclic codes over $\mathbb{Z}_{q}R,$ also we determine their minimal spanning sets and their sizes. Further, by using the Gray images of skew constacyclic codes over $\mathbb{Z}_{q}R$ we obtained some new linear codes over $\mathbb{Z}_{4}$. Finally, we have generalized these codes to double skew constacyclic codes over $\mathbb{Z}_{q}R$.
This paper considers a new alphabet set, which is a ring that we call $\mathbb{F}_4R$, to construct linear error-control codes. Skew cyclic codes over the ring are then investigated in …
This paper considers a new alphabet set, which is a ring that we call $\mathbb{F}_4R$, to construct linear error-control codes. Skew cyclic codes over the ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize $\mathbb{F}_4R$-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over $\mathbb{F}_4$ are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of $R$-skew cyclic codes which are reversible complement.