Structure and Performance of Generalized Quasi-Cyclic Codes

Güneri, Cem; Özbudak, Ferruh; Özkaya, Buket; Saçıkara, Elif; Sepasdar, Zahra; Solé, Patrick

doi:10.48550/arxiv.1702.00153

Cited by 4 publications

(12 citation statements)

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“…Proof. Working in a similar manner as in Theorem 4.2 of Güneri et al [6], the desired result follows.…”

Section: Discussionmentioning

confidence: 64%

“…In this section, we shall provide a trace description for Λ-multi-twisted codes of length n over F q by extending the work of Güneri et al [6] to Λ-multi-twisted codes. Towards this, for 1 ≤ w ≤ r and 1 ≤ i ≤ ℓ, we recall that if ǫ w,i = 1, then g w (x) divides x mi − λ i in F q [x], and the ideal x m i −λi gw(x) is a minimal λ i -constacyclic code of length m i over F q , whose generating idempotent is denoted by Θ w,i .…”

Section: Trace Description Of λ-Multi-twisted Codesmentioning

confidence: 99%

“…Generating sets of dual codes of some Λ-multi-twisted codes are expressed in terms of generating sets of the corresponding Λ-multi-twisted codes. A trace description for these codes is given and a lower bound on their minimum Hamming distances is also obtained by extending the work of Güneri et al [6] to Λ-multi-twisted codes.…”

mentioning

confidence: 94%

“…They also obtained an improved lower bound on their minimum Hamming distances. Güneri et al [6] decomposed GQC codes as direct sums of concatenated codes, which leads to a trace formula and a minimum distance bound for GQC codes. Jia [9] decomposed quasi-twisted (QT) codes and their dual codes over finite fields to a direct sum of linear codes over rings, and provided a method to construct quasi-twisted codes by using generalized discrete Fourier transform.…”

mentioning

confidence: 99%

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Multi-twisted codes over finite fields and their dual codes

Sharma

Chauhan

Singh

2018

Finite Fields and Their Applications

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Let Fq denote the finite field of order q, let m1, m2, · · · , m ℓ be positive integers satisfying gcd(mi, q) = 1 for 1 ≤ i ≤ ℓ, and let n = m1 + m2 + · · · + m ℓ . Let Λ = (λ1, λ2, · · · , λ ℓ ) be fixed, where λ1, λ2, · · · , λ ℓ are non-zero elements of Fq. In this paper, we study the algebraic structure of Λ-multi-twisted codes of length n over Fq and their dual codes with respect to the standard inner product on F n q . We provide necessary and sufficient conditions for the existence of a self-dual Λ-multi-twisted code of length n over Fq, and obtain enumeration formulae for all self-dual and self-orthogonal Λ-multi-twisted codes of length n over Fq. We also derive some sufficient conditions under which a Λ-multi-twisted code is LCD. We determine the parity-check polynomial of all Λ-multi-twisted codes of length n over Fq and obtain a BCH type bound on their minimum Hamming distances. We also determine generating sets of dual codes of some Λ-multi-twisted codes of length n over Fq from the generating sets of the codes. Besides this, we provide a trace description for all Λ-multi-twisted codes of length n over Fq by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure.Theorem and the results derived in Ling and Solé [12]. They also obtained an improved lower bound on their minimum Hamming distances. Güneri et al. [6] decomposed GQC codes as direct sums of concatenated codes, which leads to a trace formula and a minimum distance bound for GQC codes. Jia [9] decomposed quasi-twisted (QT) codes and their dual codes over finite fields to a direct sum of linear codes over rings, and provided a method to construct quasi-twisted codes by using generalized discrete Fourier transform. Saleh and Esmaeili [16] gave some sufficient conditions under which a quasi-twisted code is LCD. In a recent work, Aydin and Halilovic [1] introduced multi-twisted (MT) codes as generalization of quasi-twisted codes. They studied basic properties of 1-generator multi-twisted codes and provided a lower bound on their minimum Hamming distances. The family of multi-twisted codes is much broader as compared to quasi-twisted and constacyclic codes.Throughout this paper, let F q denote the finite field of order q, and let Λ = (λ 1 , λ 2 , · · · , λ ℓ ) be fixed, where λ 1 , λ 2 , · · · , λ ℓ are non-zero elements of F q . Let n = m 1 + m 2 + · · ·+ m ℓ , where m 1 , m 2 , · · · , m ℓ are positive integers coprime to q. The main aim of this paper is to study the algebraic structure of Λ-multi-twisted codes of length n over F q and their dual codes with respect to the standard inner product on F n q . Enumeration formulae for their two interesting subclasses, viz. self-dual and self-orthogonal codes, are also obtained. These enumeration formulae are useful in the determination of complete lists of inequivalent self-dual and self-orthogonal Λ-multitwisted codes [7, Section 9.6...

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“…Proof. Working in a similar manner as in Theorem 4.2 of Güneri et al [6], the desired result follows.…”

Section: Discussionmentioning

confidence: 64%

Section: Trace Description Of λ-Multi-twisted Codesmentioning

confidence: 99%

mentioning

confidence: 94%

mentioning

confidence: 99%

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Multi-twisted codes over finite fields and their dual codes

Sharma

Chauhan

Singh

2018

Finite Fields and Their Applications

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“…In [8], quasi-cyclic codes that are LCD have been characterized and studied using their concatenated structures. Criteria for complementary duality of generalized quasi-cyclic codes (GQC) bearing on the component codes are given and some explicit long GQC that are LCD, but not quasi-cyclic, have been exhibited in [7]. In [4], Dinh, Nguyend and Sriboonchitta investigated the algebraic structure of λ-constacyclic codes over finite commutative semi-simple rings.…”

Section: Introductionmentioning

confidence: 99%

Euclidean and Hermitian LCD MDS codes

Carlet

Mesnager

Tang

et al. 2018

Des. Codes Cryptogr.

106

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Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) are of much interest from many viewpoints due to their theoretical and practical properties. However, little work has been done on LCD MDS codes. In particular, determining the existence of q-ary [n, k] LCD MDS codes for various lengths n and dimensions k is a basic and interesting problem. In this paper, we firstly study the problem of the existence of q-ary [n, k] LCD MDS codes and completely solve it for the Euclidean case. More specifically, we show that for q > 3 there exists a q-ary [n, k] Euclidean LCD MDS code, where 0 ≤ k ≤ n ≤ q + 1, or, q = 2 m , n = q + 2 and k = 3 or q − 1. Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes.

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Do non-free LCD codes over finite commutative Frobenius rings exist?

Bhowmick

Fotue-Tabue

Martı́nez-Moro

et al. 2020

Des. Codes Cryptogr.

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In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings. We later show that a free constacyclic code over finite chain ring is LCD if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible over finite chain rings. We illustrate the minimum Lee-distance of LCD codes over some finite commutative chain rings and demonstrate the results with examples. We also got some new optimal 4 codes of different lengths which are cyclic LCD codes over 4 .

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Structure and Performance of Generalized Quasi-Cyclic Codes

Cited by 4 publications

References 0 publications

Multi-twisted codes over finite fields and their dual codes

Multi-twisted codes over finite fields and their dual codes

Euclidean and Hermitian LCD MDS codes

Do non-free LCD codes over finite commutative Frobenius rings exist?

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