1-generator Quasi-cyclic Codes Research Articles

Some Quaternary Additive Codes Outperform Linear Counterparts

The additive codes may have better parameters than linear codes. However, it is still a challenging problem to efficiently construct additive codes that outperform linear codes, especially those with greater distances than linear codes of the same lengths and dimensions. This paper focuses on constructing additive codes that outperform linear codes based on quasi-cyclic codes and combinatorial methods. Firstly, we propose a lower bound on the symplectic distance of 1-generator quasi-cyclic codes of index even. Secondly, we get many binary quasi-cyclic codes with large symplectic distances utilizing computer-supported combination and search methods, all of which correspond to good quaternary additive codes. Notably, some additive codes have greater distances than best-known quaternary linear codes in Grassl’s code table (bounds on the minimum distance of quaternary linear codes http://www.codetables.de) for the same lengths and dimensions. Moreover, employing a combinatorial approach, we partially determine the parameters of optimal quaternary additive 3.5-dimensional codes with lengths from 28 to 254. Finally, as an extension, we also construct some good additive complementary dual codes with larger distances than the best-known quaternary linear complementary dual codes in the literature.

One-generator quasi-cyclic codes and their dual codes

In this article, we study 1-generator quasi-cyclic (QC) codes over fields. QC codes of index l can be viewed as linear codes of length l. We give a description of the dual codes of 1-generator QC codes of index 2. Moreover, we obtain examples of quantum error-correcting codes (QECCs) from this study. They have better parameters than those from the online database. We consider self orthogonality conditions on such codes. We also construct some self orthogonal best known linear codes (BKLCs) linear codes, and using them we compute quantum codes. Using our symplectic study of QC codes, we are presenting three quantum codes with record-breaking parameters which improve the current record.

Quasi-cyclic codes: algebraic properties and applications

In this paper, we provide criteria for the reversibility and conjugate-reversibility of 1-generator quasi-cyclic codes. The Chinese remainder theorem is used to provide a characterization for generalized quasi-cyclic codes to be Galois linear complementary-dual and Galois self-dual. Using the approach proposed by Guneri and Ozbudak (IEEE Trans Inf Theory 59(2):979–985, 2013), a new concatenated structure for quasi-cyclic codes is given. We show that Galois linear complementary-dual quasi-cyclic codes are asymptotically good over some finite fields. In addition, DNA codes are given which have more codewords than previously known codes.

On $\sigma$ -LCD Codes

Linear complementary pairs (LCPs) of codes play an important role in armoring implementations against sidechannel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with o complementary dual (σ-LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. Like Euclidean LCD codes, σ-LCD codes can also be used to construct LCP of codes. We show that for q 2, all q-ary linear codes are σ-LCD, and for every binary linear code C, the code {0} × C is σ-LCD. Furthermore, we study deeply σ-LCD generalized quasi-cyclic (GQC) codes. In particular, we provide the characterizations of σ-LCD GQC codes, self-orthogonal GQC codes, and self-dual GQC codes, respectively. Moreover, we provide the constructions of asymptotically good σ-LCD GQC codes. Finally, we focus on σ-LCD abelian codes and prove that all abelian codes in a semisimple group algebra are σ-LCD. The results derived in this paper extend those on the classical LCD codes and show that σ-LCD codes allow the construction of LCP of codes more easily and with more flexibility.

Structure and performance of generalized quasi-cyclic codes

Generalized quasi-cyclic (GQC) codes form a natural generalization of quasi-cyclic (QC) codes. They are viewed here as mixed alphabet codes over a family of ring alphabets. Decomposing these rings into local rings by the Chinese Remainder Theorem yields a decomposition of GQC codes into a sum of concatenated codes. This decomposition leads to a trace formula, a minimum distance bound, and to a criteria for the GQC code to be self-dual or to be linear complementary dual (LCD). Explicit long GQC codes that are LCD, but not QC, are exhibited.

A Chinese remainder theorem approach to skew generalized quasi-cyclic codes over finite fields

In this work, we study a class of generalized quasi-cyclic (GQC) codes called skew GQC codes. By the factorization theory of ideals, we give the Chinese Remainder Theorem in the skew polynomial ring, which leads to a canonical decomposition of skew GQC codes. We also focus on some characteristics of skew GQC codes in details. For a 1-generator skew GQC code, we define the parity-check polynomial, determine the dimension and give a lower bound on the minimum Hamming distance. The skew QC codes are also discussed briefly.

On generator and parity-check polynomial matrices of generalized quasi-cyclic codes

Generalized quasi-cyclic (GQC) codes have been investigated as well as quasi-cyclic (QC) codes, e.g., on the construction of efficient low-density parity-check codes. While QC codes have the same length of cyclic intervals, GQC codes have different lengths of cyclic intervals. Similarly to QC codes, each GQC code can be described by an upper triangular generator polynomial matrix, from which the systematic encoder is constructed. In this paper, a complete theory of generator polynomial matrices of GQC codes, including a relation formula between generator polynomial matrices and parity-check polynomial matrices through their equations, is provided. This relation generalizes those of cyclic codes and QC codes. While the previous researches on GQC codes are mainly concerned with 1-generator case or linear algebraic approach, our argument covers the general case and shows the complete analogy of QC case. We do not use Gröbner basis theory explicitly in order that all arguments of this paper are self-contained. Numerical examples are attached to the dual procedure that extracts one from each other. Finally, we provide an efficient algorithm which calculates all generator polynomial matrices with given cyclic intervals.

A class of 1-generator quasi-cyclic codes over finite chain rings

Using another method, which is different with the one given by Cao [1-generator quasi-cyclic (QC) codes over finite chain rings, Appl. Algebra Eng. Commun. Comput. 24 (2013), pp. 53–72], we investigate the structural properties of a class of 1-generator QC codes over finite chain rings. We give the structure of the annihilator of 1-generator QC codes, the conditions for 1-generator QC codes to be free and the minimum distance bounds on 1-generator QC codes. Under some conditions, we also 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.

Quasi-abelian codes

We study $$H$$ H -quasi-abelian codes in $$\mathbb F _q[G]$$ F q [ G ] , where $$H\le G$$ H ≤ G are abelian groups such that $$\gcd (|H|,q)=1$$ gcd ( | H | , q ) = 1 . Such codes are generalizations of quasi-cyclic codes and can be viewed as linear codes over the group ring $$\mathbb F _q[H]$$ F q [ H ] . Using the Discrete Fourier Transform, $$\mathbb F _q[H]$$ F q [ H ] can be decomposed as a direct product of finite fields. This decomposition leads us to a structural characterization of quasi-abelian codes and their duals. Necessary and sufficient conditions for such codes to be self-dual are given together with the enumeration based on $$q$$ q -cyclotomic classes of $$H$$ H . In particular, when $$H$$ H is an elementary $$p$$ p -group, we characterize the $$q$$ q -cyclotomic classes of $$H$$ H and give an explicit formula for the number of self-dual $$H$$ H -quasi abelian codes. Analogous to 1-generator quasi-cyclic codes, we investigate the structural characterization and enumeration of 1-generator quasi-abelian codes. We show that the class of binary self-dual (strictly) quasi-abelian codes is asymptotically good. Finally, we present four strictly quasi-abelian codes and ten codes obtained by puncturing and shortening of these codes, whose minimum distances are better than the lower bound in Grassl's online table.

1-Generator quasi-cyclic codes over [formula omitted

In this paper, we mainly consider the quasi-cyclic (QC) codes over finite chain ring R=Fpm+uFpm+⋯+us−1Fpm, where p is a prime number and m, s are positive integers such that s≥2 and us=0. We investigate the structural properties of the QC codes over R, regarding them as subcodes of cyclic codes over some Galois extension rings of R. This point of view leads to construct the 1-generator QC codes with index s over finite field Fpm. Further, we study the structural properties of annihilators of the 1-generator QC codes. For the case s=2, under the conditions gcd(n,p)=1 and gcd(|pm|n,l)=1, we discuss the enumeration of the distinct 1-generator QC codes and describe how to obtain one and the only one generator for each 1-generator QC code. Finally, we give some examples to illustrate the main work in this paper.

1-generator quasi-cyclic codes over finite chain rings

Let R be an arbitrary commutative finite chain ring with \(1\ne 0\). 1-generator quasi-cyclic (QC) codes over R are considered in this paper. Let \(\gamma \) be a fixed generator of the maximal ideal of R, \(F=R/\langle \gamma \rangle \) and \(|F|=q\). For any positive integers m, n satisfying \(\mathrm{gcd}(q,n)=1\), let \(\mathcal{R}_n=R[x]/\langle x^n-1\rangle \). Then 1-generator QC codes over R of length mn and index m can be regarded as 1-generator \(\mathcal{R}_n\)-submodules of the module \(\mathcal{R}_n^m\). First, we consider the parity check polynomial of a 1-generator QC code and the properties of the code determined by the parity check polynomial. Then we give the enumeration of 1-generator QC codes with a fixed parity check polynomial in standard form over R. Finally, under the condition that \(\mathrm{gcd}(|q|_n,m)=1\), where \(|q|_n\) denotes the order of q modulo n, we describe an algorithm to list all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial in standard form over R of length mn and index m.

A class of 1-generator repeated root quasi-cyclic codes

Let $$q$$ be a power of a prime integer $$p, m=p^em_0$$ and $$|q|_{m_{0}}$$ the order of $$q$$ modulo $$m_0$$ . By use of finite commutative chain ring theory, an algorithm to construct all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial over a finite field $$F_q$$ of length $$mn$$ and index $$n$$ , under the condition that $$\mathrm {gcd}(|q|_{m_0},n)=1$$ , are given.

Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration

For R a Galois ring and m 1, . . . , m l positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m 1, m 2, . . . , m l ) and length $${\sum_{i=1}^lm_i}$$ is an R[x]-submodule of $${R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}$$. Suppose m 1, . . . , m l are all coprime to the characteristic of R and let {g 1, . . . , g t } be the set of all monic basic irreducible polynomials in the factorizations of $${x^{m_i}-1}$$ (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m 1, m 2, . . . , m l ) and length $${\sum_{i=1}^lm_i}$$ are identified with $${{\mathcal G}_1\times\cdots\times {\mathcal G}_t}$$, where $${{\mathcal G}_j}$$ is an R[x]/(g j )-submodule of $${(R[x]/(g_j))^{n_j}}$$, where n j is the number of i for which g j appears in the factorization of $${x^{m_i}-1}$$ into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths.

Quaternary 1-generator quasi-cyclic codes

Quaternary 1-generator quasi-cyclic codes are considered in the paper. Under the conditions that n is odd and gcd(|2| n , m) = 1, where |2| n denotes the order of 2 modulo n, we give the enumeration of quaternary 1-generator quasi-cyclic codes of length mn, and describe an algorithm which will obtain one, and only one, generator for each quaternary 1-generator quasi-cyclic code.

On quasi-cyclic codes over $${\mathbb{Z}_q}$$

Quasi-cyclic (QC) codes are a remarkable generalization of cyclic codes. Many QC codes have been shown to be best for their parameters. In this paper, some structural properties of QC codes over the prime power integer residue ring $${\mathbb{Z}_q}$$are considered. An l-QC code of length lm over $${\mathbb{Z}_q}$$is viewed both as in the conventional row circulant form and also as a $${\frac{\mathbb{Z}_q[x]}{\langle x^m-1 \rangle}}$$-submodule of $${\frac{GR(q,l)[x]}{\langle x^m-1 \rangle}}$$, where GR(q, l) is the Galois extension ring of degree l over $${\mathbb{Z}_q}$$. A necessary and sufficient condition for cyclic codes over Galois rings to be free is obtained and a BCH type bound for them is also given. A sufficient condition for 1-generator QC codes to be $${\mathbb{Z}_q}$$-free is given and a formula to evaluate their ranks is derived. Some distance bounds for 1-generator QC codes are also discussed. The duals of QC codes over $${\mathbb{Z}_q}$$are also briefly discussed.

Generalized quasi-cyclic codes: structural properties and code construction

Generalized quasi-cyclic (GQC) codes are defined by generator matrices comprised of circulant matrices of lengths not necessarily identical. A decomposition of these codes is given by using the Chinese reminder theorem. The focus is to characterize ρ-generator GQC codes in details. A good lower bound on the minimum distance of such a code in terms of the minimum distance of the constituent codes is given. Construction methods are given and a set of GQC codes is provided that from minimum distance perspective are optimal codes among the known linear codes having the same length and dimension.

On complementary-dual quasi-cyclic codes

A linear code C is said to be a complementary-dual code ( an LCD code) if it satisfies C ∩ C ⊥ = { 0 } where C ⊥ is the dual of C. This paper is to identify few classes of LCD quasi-cyclic (QC) codes. A sufficient condition for a ρ-generator QC code C is given under which C is an LCD code. Another sufficient condition is given for maximal 1-generator codes. We provide two necessary and sufficient conditions for a maximal 1-generator QC code C satisfying certain constraints to be an LCD code. It is shown that unlike cyclic codes, a maximal 1-generator index-2 QC code is reversible if and only if it is a self-dual code. Several classes of LCD QC codes are introduced.

Computation of Groebner Basis for Systematic Encoding of Generalized Quasi-Cyclic Codes

Generalized quasi-cyclic (GQC) codes form a wide and useful class of linear codes that includes thoroughly quasi-cyclic codes, finite geometry (FG) low density parity check (LDPC) codes, and Hermitian codes. Although it is known that the systematic encoding of GQC codes is equivalent to the division algorithm in the theory of Grobner basis of modules, there has been no algorithm that computes Grobner basis for all types of GQC codes. In this paper, we propose two algorithms to compute Grobner basis for GQC codes from their parity check matrices: echelon canonical form algorithm and transpose algorithm. Both algorithms require sufficiently small number of finite-field operations with the order of the third power of code-length. Each algorithm has its own characteristic; the first algorithm is composed of elementary methods, and the second algorithm is based on a novel formula and is faster than the first one for high-rate codes. Moreover, we show that a serial-in serial-out encoder architecture for FG LDPC codes is composed of linear feedback shift registers with the size of the linear order of code-length; to encode a binary codeword of length n, it takes less than 2n adder and 2n memory elements. Keywords: automorphism group, Buchberger's algorithm, division algorithm, circulant matrix, finite geometry low density parity check (LDPC) codes.

1-Generator Quasi-Cyclic Codes

This paper discusses the enumeration of 1-generator quasi-cyclic codes and describes an algorithm which will obtain one, and only one, generator for each 1-generator quasi-cyclic code.

New Quasi-Cyclic Codes From Simplex Codes

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