Further designs for self-orthogonal and LCD codes developed from functions over finite fields

A linear complementary-dual (LCD) code C is a linear code whose dual code $$C^{\perp }$$ satisfies $$C \cap C^{\perp }=\{0\}$$ . In this work we characterize some classes of LCD q-ary $$(\lambda , l)$$ -quasi-twisted (QT) codes of length $$n=ml$$ with $$(m,q)=1$$ , $$\lambda \in F_{q} \setminus \{0\}$$ and $$\lambda \ne \lambda ^{-1}$$ . We show that every $$(\lambda ,l)$$ -QT code C of length $$n=ml$$ with $$dim(C)<m$$ or $$dim(C^{\perp })<m$$ is an LCD code. A sufficient condition for r-generator QT codes is provided under which they are LCD. We show that every maximal 1-generator $$(\lambda ,l)$$ -QT code of length $$n=ml$$ with $$l>2$$ is either an LCD code or a self-orthogonal code and a sufficient condition for this family of codes is given under which such a code C is LCD. Also it is shown that every maximal 1-generator $$(\lambda ,2)$$ -QT code is LCD. Several good and optimal LCD QT codes are presented.

Linear complementary dual (LCD) codes is a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes have been employed in cryptography. More specifically, it has been shown that LCD codes can also help improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault non-invasive attacks. In this paper, we are interested in the construction of particular algebraic geometry (AG) LCD codes which could be good candidates to be resistant against SCA. We firstly provide a construction scheme for obtaining LCD codes from elliptic curves. Then, some explicit LCD codes from elliptic curve are presented. MDS codes are of the most importance in coding theory due to their theoretical significance and practical interests. In this paper, all the constructed LCD codes from elliptic curves are MDS or almost MDS. Some infinite classes of LCD codes from elliptic curves are optimal due to the Griesmer bound. Finally, we introduce a construction mechanism for obtaining LCD codes from any algebraic curve and derive some explicit LCD codes from hyperelliptic curves and Hermitian curves.

Due to their important applications in theory and practice, linear complementary dual (LCD) codes and self-orthogonal codes have received much attention in the last decade. The objective of this paper is to extend a recent construction of binary LCD codes and self-orthogonal codes to the general $ p $-ary case, where $ p $ is an odd prime. Based on the extended construction, several classes of $ p $-ary linear codes are obtained. The characterizations of these linear codes to be LCD or self-orthogonal are derived. The duals of these linear codes are also studied. It turns out that the proposed linear codes are optimal in many cases in the sense that their parameters meet certain bounds on linear codes. The weight distributions of these linear codes are settled.

Linear complementary dual (LCD) codes are linear codes satisfying . Researchers have proved to construct linear codes over finite fields , q>3, is equivalent to construct LCD codes. This means that the investigation of binary and ternary LCD codes is the only remaining open problem. In this work, we propose new constructions of binary and ternary LCD codes by using matrix-product codes and prove that binary and ternary LCD codes from matrix-product codes are asymptotically good. Then we construct some new and good binary and ternary LCD codes using matrix-product matrices.

Linear codes with certain special properties have received renewed attention in recent years due to their practical applications. Among them, binary linear complementary dual (LCD) codes play an important role in implementations against side-channel attacks and fault injection attacks. Self-orthogonal codes can be used to construct quantum codes. In this paper, four classes of binary linear codes are constructed via a generic construction which has been intensively investigated in the past decade. Simple characterizations of these linear codes to be LCD or self-orthogonal are presented. Resultantly, infinite families of binary LCD codes and self-orthogonal codes are obtained. Infinite families of binary LCD codes from the duals of these four classes of linear codes are produced. Many LCD codes and self-orthogonal codes obtained in this paper are optimal or almost optimal in the sense that they meet certain bounds on general linear codes. In addition, the weight distributions of two sub-families of the proposed linear codes are established in terms of Krawtchouk polynomials.

Further results on Euclidean and Hermitian linear complementary dual codes

No AccessRepeated-root bidimensional (μ, ν)-constacyclic codes of length 4pt.2rShikha Patel and Om PrakashShikha PatelDepartment of Mathematics, Indian Institute of Technology Patna, Patna-801106, India and Om PrakashDepartment of Mathematics, Indian Institute of Technology Patna, Patna-801106, IndiaPublished Online:October 13, 2020pp 266-289https://doi.org/10.1504/IJICOT.2020.110738PDF ToolsAdd to FavouritesDownload CitationsTrack Citations Share this article on social mediaShareShare onFacebookTwitterLinkedInReddit AboutAbstractLet p be an odd prime. The main concern of this article is to study all the repeated-root bidimensional (μ, ν)-constacyclic codes of length 4pt.2r over the finite field 𝔽pm. Here, we provide all the self-dual repeated-root bidimensional (1, 1)-constacyclic and (−1, 1)-constacyclic codes of length 4pt.2r over 𝔽pm. We also discuss the repeated-root bidimensional (η, 1)-constacyclic codes of length 4pt.2r over 𝔽pm. Moreover, it has been shown that these structures are useful in the construction of linear complementary dual (LCD) codes and self-dual codes. As an example, we are listed all the repeated-root bidimensional (μ, ν)-constacyclic codes of length 72 over the finite field 𝔽27.Keywordscyclic codes, constacyclic codes, two-dimensional constacyclic codes, dual codes, LCD codes, repeated-root codes Previous article Next article FiguresReferencesRelatedDetails Volume 5Issue 3-42020 ISSN: 1753-7703eISSN: 1753-7711 HistoryPublished onlineOctober 13, 2020 Copyright © 2020 Inderscience Enterprises Ltd.Keywordscyclic codesconstacyclic codestwo-dimensional constacyclic codesdual codesLCD codesrepeated-root codesAuthors and AffiliationsShikha Patel1 Om Prakash2 1. Department of Mathematics, Indian Institute of Technology Patna, Patna-801106, India2. Department of Mathematics, Indian Institute of Technology Patna, Patna-801106, IndiaPDF download

Linear complementary dual (LCD) codes are linear codes whose intersections with their duals are trivial. In this paper, characterizations of LCD codes with respect to the symplectic inner product, i.e. symplectic LCD codes, over finite fields are given. Some methods for constructing symplectic LCD codes and symplectic LCD MDS codes are presented. As an application, a class of symplectic LCD MDS codes is constructed by employing Vandermonde matrices, and the corresponding MDS maximal entanglement entanglement-assisted quantum error-correcting codes (EAQECCs) are constructed.

Theory of additive complementary dual codes, constructions and computations

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.

Let $$\mathbb {F}_q$$ be a finite field with q elements, $$D_{2n,\,r}$$ a generalized dihedral group with $$\gcd (2n,q)=1$$ , and $$\mathbb {F}_q[D_{2n,\,r}]$$ a generalized dihedral group algebra. Firstly, an explicit expression for primitive idempotents of $$\mathbb {F}_q[D_{2n,\,r}]$$ is determined, which extends the results of Brochero Martinez (Finite Fields Appl 35:204–214, 2015). Secondly, all linear complementary dual (LCD) codes and self-orthogonal codes in $$\mathbb {F}_q[D_{2n,\,r}]$$ are precisely described and counted. Some numerical examples are also presented to illustrate our main results.

Cyclic codes over a non-chain ring Re,q and their application to LCD codes

Linear complementary dual (LCD) cyclic codes were referred historically to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their symplectic basis. Using such a characterization,we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results of the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of q-ary LCD codes have orthonormal basis, where q is a power of an odd prime.

On the minimum weights of binary LCD codes and ternary LCD codes

It was shown by Massey that linear complementary dual (LCD) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert–Varshamov (GV) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound.