Based on cyclic simplex codes, a new construction of a family of 2-generator quasi-cyclic two-weight codes is given. New optimal binary quasi-cyclic [195, 8, 96], [210, 8, 104] and [240, 8, 120] codes, good QC ternary [195, 6, 126], [208, 6, 135], [221, 6, 144] codes are thus obtained. Furthermre, binary quasi-cyclic self-complementary codes are also constructed.
We generalize Sepasdar's method for finding a generator matrix of two-dimensional cyclic codes to find an independent subset of a general multicyclic code, which may form a basis of the code as a vector subspace. A generator matrix can be then constructed from this basis.
In this paper, we study the structure of 1-generator quasi-cyclic codes over the ring R = F2 + uF2 + vF2 + uvF2, with u2 = v2 = 0 and uv = vu. We determine the minimal spanning sets for these codes. As a generalization of these codes, we also investigate the structure of 1-generator generalized quasi-cyclic codes over R and determine a BCH type bound for them.
Quasi-cyclic (QC) codes form an important generalization of cyclic codes. It is well know that QC codes of length $s\ell$ with index $s$ over the finite field $\mathbb{F}$ are $\mathbb{F}[y]$-submodules of the ring $\frac{\mathbb{F}[x,y]}{< x^s-1,y^{\ell}-1 >}$. The aim of the present paper, is to study QC codes of length $s\ell$ with index $s$ over the finite field $\mathbb{F}$ and find generator polynomials and generator matrix for these codes. To achieve this aim, we apply a novel method to find generator polynomials for $\mathbb{F}[y]$-submodules of $\frac{\mathbb{F}[x,y]}{< x^s-1,y^{\ell}…
The aim of this paper is to determine the algebraic structure of multidimensional cyclic codes over a finite chain ring $\mathfrak{R}$. An algorithm to find the generator polynomials of $n$ dimensional ($n$D) cyclic codes of length $m_{1}m_{2}\dots m_{n}$ over $\mathfrak{R}$ has been developed using the generator polynomials of cyclic codes over $\mathfrak{R}$. Additionally, the generators of $n$D cyclic codes with length $m_{1}m_{2}\dots m_{n}$ over $\mathfrak{R}$ have been obtained as separable polynomials for the case $q\equiv 1(mod~ m_{j}), j\geq 2$, where $q=p^{r}$ is the cardinality of r…
As a generalization of cyclic codes, quasi-cyclic (QC) codes contain many good linear codes. But quasi-cyclic codes studied so far are mainly limited to one generator (1-generator) QC codes. In this correspondence, 2-generator and 3-generator QC codes are studied, and many good, new QC codes are constructed from simplex codes. Some new binary QC codes or related codes, that improve the bounds on maximum minimum distance for binary linear codes are constructed. They are 5-generator QC [93, 17, 34] and [254, 23, 102] codes, and related [96, 17, 36], [256, 23, 104] codes.
We present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes, that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error flo…
In this paper, we mainly consider quasi-cyclic (QC) codes over finite chain rings. We study module structures and trace representations of QC codes, which lead to some lower bounds on the minimum Hamming distance of QC codes. Moreover, we investigate the structural properties of 1-generator QC codes. Under some conditions, we discuss the enumeration of 1-generator QC codes and describe how to obtain the one and only one generator for each 1-generator QC code.
In this article, we study skew cyclic codes over ring $R=\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}$, where $q=p^{m}$, $p$ is an odd prime and $v^{3}=v$. We describe generator polynomials of skew cyclic codes over this ring and investigate the structural properties of skew cyclic codes over $R$ by a decomposition theorem. We also describe the generator polynomials of the duals of skew cyclic codes. Moreover, the idempotent generators of skew cyclic codes over $\mathbb{F}_{q}$ and $R$ are considered.
Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of $p$-ary cyclic codes whose duals have three zeros are proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.
A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code ${\cal C}\subseteq{\mathbb{Z}}_2^α\times{\mathbb{Z}}_4^β$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of ${\mathbb{Z}}_4$ coordinates, such that any simultaneous cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the $\mathbb{Z}_4[x]$-module $\mathbb{Z}_2[x]/(x^α-1)\times\mathbb{Z}_4[x]/(x^β-1)$. Any $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic code ${\cal C}$ is of the form $\langle (b(x)\mid{ 0}), (\ell…
In this paper we study Z2Z4Z8-additive codes, which are the extension of recently introduced Z2Z4-additive codes. We determine the standard forms of the generator and parity-check matrices of Z2Z4Z8-additive codes. Moreover, we investigate Z2Z4Z8-cyclic codes giving their generator polynomials and spanning sets. We also give some illustrative examples of both Z2Z4Z8-additive codes and Z2Z4Z8-cyclic codes.
A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code ${\cal C}\subseteq{\mathbb{Z}}_2^α\times{\mathbb{Z}}_4^β$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of ${\mathbb{Z}}_4$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. These codes can be identified as submodules of the $\mathbb{Z}_4[x]$-module $\mathbb{Z}_2[x]/(x^α-1)\times\mathbb{Z}_4[x]/(x^β-1)$. The parameters of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic code are stated in terms of the degrees of the generat…
Coding theory and combinatorial $t$-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some $t$-designs have been constructed with this approach by using certain linear codes in recent years. However, only a few infinite families of cyclic codes holding an infinite family of $3$-designs are reported in the literature. The objective of this paper is to study an infinite family of cyclic codes and determine their parameters. By the parameters of these co…
The generalized Hamming weights (GHWs) are fundamental parameters of linear codes. GHWs are of great interest in many applications since they convey detailed information of linear codes. In this paper, we continue the work of [10] to study the GHWs of a family of cyclic codes with arbitrary number of nonzeroes. The weight hierarchy is determined by employing a number-theoretic approach.
Let $q$ be a power of a prime $p$. In this paper, we study reversible cyclic codes of arbitrary length over the ring $ R = \mathbb{F}_q + u \mathbb{F}_q$, where $u^2=0 mod q$. First, we find a unique set of generators for cyclic codes over $R$, followed by a classification of reversible cyclic codes with respect to their generators. Also, under certain conditions, it is shown that dual of reversible cyclic code is reversible over $\mathbb{Z}_2+u\mathbb{Z}_2$. Further, to show the importance of these results, some examples of reversible cyclic codes are provided.
Let m, k be positive integers, p be an odd prime and $π$ be a primitive element of $\mathbb{F}_{p^m}$. In this paper, we determine the weight distribution of a family of cyclic codes $\mathcal{C}_t$ over $\mathbb{F}_p$, whose duals have two zeros $π^{-t}$ and $-π^{-t}$, where $t$ satisfies $t\equiv \frac{p^k+1}{2}p^τ\ ({\rm mod}\ \frac{p^m-1}{2}) $ for some $τ\in \{0,1,\cdots, m-1\}$.
Quasi-twisted (QT) codes are a generalization of quasi-cyclic (QC) codes. Based on consta-cyclic simplex codes, a new explicit construction of a family of 2-generator quasi-twisted (QT) two-weight codes is presented. It is also shown that many codes in the family meet the Griesmer bound and therefore are length-optimal. New distance-optimal binary QC [195, 8, 96], [210, 8, 104] and [240, 8, 120] codes, and good ternary QC [208, 6, 135] and [221, 6, 144] codes are also obtained by the construction.
Based on cyclic and consta-cyclic simplex codes, a new explicit construction of a family of two-weight codes is presented. These two-weight codes obtained are in the form of 2-generator quasi-cyclic, or quasi-twisted structure. Based on this construction, new optimal binary quasi-cyclic [195, 8, 96], [210, 8, 104] and [240, 8, 120] codes, and good QC ternary [208, 6, 135] and [221, 6, 144] codes are thus obtained. It is also shown that many codes among the family meet the Griesmer bound and thereful are optimal.
Ovoids in $\PG(3, q)$ have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families are known. The first is the elliptic quadratics and the second is the Tits ovoids. In this article, we present a family of ovoids in $\PG(3, 2^m)$ for all $m$ which are from a family of irreducible cyclic codes.
We characterize the affine-invariant maximal extended cyclic codes. Then by the CSS construction, we derive from these codes a family of pure quantum codes. Also for ordnq even, a new family of degenerate quantum stabilizer codes is derived from the classical duadic codes. This answer an open problem asked by Aly et al.
We present a family of reducible cyclic codes constructed as the direct sum of two different semiprimitive two-weight irreducible cyclic codes. This family generalizes the class of reducible cyclic codes that was reported in the main result of B. Wang, {\em et al.} \cite{once}. Moreover, despite of what was stated therein, we show that, at least for the codes studied here, it is still possible to compute the frequencies of their weight distributions through the cyclotomic numbers in a very easy way.
A Z2-triple cyclic code of block length (r,s,t) is a binary code of length r+s+t such that the code is partitioned into three parts of lengthsr,s andt such that each of the three parts is invariant under the cyclic shifts of the coordinates. Such a code can be viewed as Z2[x]-submodules of Z_2[x]/<x^r-1>xZ_2[x]/<x^s-1>xZ_2[x]/<x^t-1>, in polynomial representation. In this paper, we determine the structure of these codes. We have obtained the form of the generators for such codes. Further, a minimal generating set for such a code is obtained. Also, we study the structure of the duals of these c…
We give a polynomial representation for additive cyclic codes over $\mathbb{F}_{p^2}$. This representation will be applied to uniquely present each additive cyclic code by at most two generator polynomials. We determine the generator polynomials of all different additive cyclic codes. A minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic codes over $\mathbb{F}_p$. We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal additive cyclic codes over $\mathbb{F}_{p^2}$. Finally, we present ten record-breaking binary quantum …
Cyclic codes have wide applications in data storage systems and communication systems. Employing two-prime Whiteman generalized cyclotomic sequences of order 6, we construct several classes of cyclic codes over the finite field GF}(q) and give their generator polynomials. And we also calculate the minimum distance of some cyclic codes and give lower bounds of the minimum distance for some other cyclic codes.
Cyclic codes are an important subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics. In this paper, two families of optimal ternary cyclic codes are presented. The first family of cyclic codes has parameters $[3^m-1, 3^m-1-2m, 4]$ and contains a class of conjectured cyclic codes and several new classes of optimal cyclic codes. The second family of cyclic codes has parameters $[3^m-1, 3^m-2-2m, 5]$ and contains a number of classes of cyclic codes that are obtained from perfect nonlinear functions over $\fthreem$, where $m>1$ …
In this work, a unique set of generators for a cyclic code over a finite chain ring has been established. The minimal spanning set and rank of the code have also been determined. Further, sufficient as well as necessary conditions for a cyclic code to be an MDS code and for a cyclic code to be an MHDR code have been obtained. Some examples of optimal cyclic codes have also been presented.
Orbit codes are a family of codes employable for communications on a random linear network coding channel. The paper focuses on the classification of these codes. We start by classifying the conjugacy classes of cyclic subgroups of the general linear group. As a result, we are able to focus the study of cyclic orbit codes to a restricted family of them.
In this work, cyclic-skew-cyclic codes and sum-rank BCH codes are introduced. Cyclic-skew-cyclic codes are characterized as left ideals of a suitable non-commutative finite ring, constructed using skew polynomials on top of polynomials (or vice versa). Single generators of such left ideals are found, and they are used to construct generator matrices of the corresponding codes. The notion of defining set is introduced, using pairs of roots of skew polynomials on top of poynomials. A lower bound (called sum-rank BCH bound) on the minimum sum-rank distance is given for cyclic-skew-cyclic codes wh…
A binary linear code $C$ is a $\mathbb{Z}_2$-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the $\mathbb{Z}_2[x]$-module $\mathbb{Z}_2[x]/(x^r-1)\times\mathbb{Z}_2[x]/(x^s-1).$ We determine the structure of $\mathbb{Z}_2$-double cyclic codes giving the generator polynomials of these codes. The related polynomial representation of $\mathbb{Z}_2$-double cyclic codes and its duals, and the relations between the polynomial generat…
A Z2Z4-additive code C subset of Z_2^alpha x Z_4^beta is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z_2 and the set of Z_4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Phi(C) be the binary Gray image of C. We study the rank and the dimension of the kernel of a Z2Z4-additive cyclic code C, that is, the dimensions of the binary linear codes <Phi(C)> and ker(Phi(C)). We give upper and lower bounds for these parameters. It is known that the codes <Phi(C)> and ker(Phi(C)) are binary images of Z…
We give a complete list of orbit codes that are generated by an irreducible cyclic group, i.e. an irreducible group having one generator. We derive some of the basic properties of these codes such as the cardinality and the minimum distance.
This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the row- and the column-code. Moreover, we provide a new lower bound on the minimum Hamming distance of a given quasi-cyclic code and present a new algebraic decoding algorithm.More specifically, we prove an explicit (unreduced) basis of an l\_a l\_b-quasi-cyclic product code in terms of the generator matrix in reduced Gr{ö}bner basis with respect to the position-…
Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on special polynomials over finite fields. The first family of linear codes are extended primitive cyclic codes which are affine-invariant. The second family of linear codes are reducible cyclic codes. The parameters of these codes and their duals are determined. As the first application, we prove that these two families of linear codes hold $t$-designs, where $…
Basic properties of a characteristic matrix for a tail-biting convolutional code are investigated. A tail-biting convolutional code can be regarded as a linear block code. Since the corresponding scalar generator matrix Gt has a kind of cyclic structure, an associated characteristic matrix also has a cyclic structure, from which basic properties of a characteristic matrix are obtained. Next, using the derived results, we discuss the possibility of trellis reduction for a given tail-biting convolutional code. There are cases where we can find a scalar generator matrix Gs equivalent to Gt based …
For odd length $n$, the cyclic codes construction over $\Re= \Z_4[v]/ \langle v^2-v \rangle$ is provided. The hulls of cyclic codes over $\Re$ are studied. The average $2$-dimension $E(n)$ of the hulls of cyclic codes over $\Re$ is also conferred. Among these, the various examples of generators of hulls of cyclic codes over $\Re$ are provided, whose $\Z_4$-images are good $\Z_4$-linear codes with good parameters.
In this article, we study the skew cyclic codes over R_{k}=F_{p}+uF_{p}+\dots +u^{k-1}F_{p} of length n. We characterize the skew cyclic codes of length $n$ over R_{k} as free left R_{k}[x;θ]-submodules of R_{k}[x;θ]/\langle x^{n}-1\rangle and construct their generators and minimal generating sets. Also, an algorithm has been provided to encode and decode these skew cyclic codes.
Let $p$ be a prime number. In this paper, we discuss the structures of cyclic codes over the ring $ \mathbb{F}_p[u, v] / \langle u^k, v^2, uv-vu\rangle$. We find a unique set of generators for these codes. We also study the rank and the Hamming distance of these codes.
The purpose of this paper is to study the cyclic self orthogonal codes over $\mathbb{Z}_{p^m}$. After providing the generator polynomial of cyclic self orthogonal codes over $\mathbb{Z}_{p^m}$, we give the necessary and sufficient condition for the existence of non-trivial self orthogonal codes over $\mathbb{Z}_{p^m}$ . We have also provided the number of such codes of length $n$ over $\mathbb{Z}_{p^m}$ for any $ (p,n) = 1 $.
In this paper, a family of cyclic codes over $\mathbb{F}_{p}$ whose duals have five zeros is presented, where $p$ is an odd prime. Furthermore, the weight distributions of these cyclic codes are determined.
Interplay between coding theory and combinatorial $t$-designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of $3$-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of $3$-designs has been reported in the literature. This is the main motivation of this paper. Let $q=p^m$, where $p$ is an odd prime and $m \geq 2$ is an integer. The objective of this paper is to present an infinite family of cyclic codes over $\gf(q)…
Given a prime power $q$ and a positive integer $r>1$ we say that a cyclic code of length $n$, $C\subseteq F_{q^r}^n$, is Galois supplemented if for any non-trivial element $σ$ in the Galois group of the extension $ F_{q^r}/ F_q$, $C+C^σ= F_{q^r}^n$, where $C^σ=\{(x_1^σ,\dots,x_n^σ)\mid (x_1,\dots,x_n)\in C\}$. This family includes the quadratic-residue (QR) codes over $ F_{q^2}$. Some important properties QR-codes are then extended to Galois supplemented codes and a new one is also considered, which is actually the motivation for the introduction of this family of codes: in a Galois supplement…
Steiner systems are a fascinating topic of combinatorics. The most studied Steiner systems are $S(2, 3, v)$ (Steiner triple systems), $S(3, 4, v)$ (Steiner quadruple systems), and $S(2, 4, v)$. There are a few infinite families of Steiner systems $S(2, 4, v)$ in the literature. The objective of this paper is to present an infinite family of Steiner systems $S(2, 4, 2^m)$ for all $m \equiv 2 \pmod{4} \geq 6$ from cyclic codes. This may be the first coding-theoretic construction of an infinite family of Steiner systems $S(2, 4, v)$. As a by-product, many infinite families of $2$-designs are also…
We investigate the notion of cyclicity for convolutional codes as it has been introduced by Piret and Roos in the seventies. Codes of this type are described as submodules of the module of all vector polynomials in one variable with some additional generalized cyclic structure but also as specific left ideals in a skew polynomial ring. Extending a result of Piret, we show in a purely algebraic setting that these ideals are always principal. This leads to the notion of a generator polynomial just like for cyclic block codes. Similarly a control polynomial can be introduced by considering the ri…
In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic ones such as generator polynomials and ideals. Indeed we show a one-to-one correspondence between l-quasi-cyclic codes of length m and ideals of M_l(Fq)[X]/(X^m-1). This permits to construct new classes of codes, namely quasi-BCH and quasi-evaluation codes. We study the parameters of such codes and propose a decoding algorithm up to half the designed minimum distance. We even found one new quasi-cyclic code with better parameters than known [189, 11, 125]_F4 and 48 der…
In this paper, we study some repeated-root two-dimensional cyclic and constacyclic codes over a finite field $F=\mathbb{F}_q$. We obtain the generator matrices and generator polynomials of these codes and their duals. We also investigate when such codes are self-dual. Moreover, we prove that if there exists an asymptotically good family of one-sided repeated-root two-dimensional cyclic or constacyclic codes, then there exists an asymptotically good family of simple root two-dimensional cyclic or constacyclic codes with parameters at least as good as the first family. Furthermore, we show that …
In this paper, a family of reducible cyclic codes over GF(p) whose duals have four zeros is presented, where p is an odd prime. Furthermore, the weight distribution of these cyclic codes is determined.
