Linear Complementary Dual Codes Research Articles

Binary and ternary leading-systematic LCD codes from special functions

Linear complementary dual codes (LCD codes for short) are an important subclass of linear codes which have nice applications in communication systems, cryptography, consumer electronics and information protection. In the literature, it has been proved that an [n,k,d] Euclidean LCD code over Fq with q>3 exists if there is an [n,k,d] linear code over Fq, where q is a prime power. However, the existence of binary and ternary Euclidean LCD codes has not been totally characterized. Hence it is interesting to construct binary and ternary Euclidean LCD codes with new parameters. In this paper, we construct new families of binary and ternary leading-systematic Euclidean LCD codes from some special functions including semibent functions, quadratic functions, almost bent functions, and planar functions. These LCD codes are not constructed directly from such functions, but come from some self-orthogonal codes constructed with such functions. Compared with known binary and ternary LCD codes, the LCD codes in this paper have new parameters.

Optimal Quaternary Hermitian LCD Codes.

Linear complementary dual (LCD) codes, which are a class of linear codes introduced by Massey, have been extensively studied in the literature recently. It has been shown that LCD codes play a role in measures to counter passive and active side-channel analyses on embedded cryptosystems. In this paper, tables are presented of good quaternary Hermitian LCD codes and they are used in the construction of puncturing, shortening and combination codes. The results of this, including three tables of the best-known quaternary Hermitian LCD codes of any length n≤25 with corresponding dimension k, are presented. In addition, many of these quaternary Hermitian LCD codes given in this paper are optimal and saturate the lower or upper bound of Grassl's code table, and some of them are nearly optimal.

Cyclic Codes over a Non-Local Non-Unital Ring

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H=⟨a,b∣2a=2b=0,a2=0,b2=b,ab=ba=0⟩. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2n over F2 is studied.

Linear complementary dual codes and double circulant codes over a semi-local ring

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> be an odd prime power and <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_q $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M3">\begin{document}$ q $\end{document}</tex-math></inline-formula> elements. In this paper, suppose ring <inline-formula><tex-math id="M4">\begin{document}$ R = \mathbb{F}_{q}+ \mu \mathbb{F}_{q}+ \nu \mathbb{F}_{q}+ \mu \nu \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \mu \nu = \nu \mu, \mu^{2} = \mu, \nu^{2} = \nu. $\end{document}</tex-math></inline-formula> We first give a Gray map from <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> onto <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_q^{4} $\end{document}</tex-math></inline-formula> and consider a decomposition of the ring <inline-formula><tex-math id="M8">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Additionally, we investigate linear complementary dual (LCD) codes over the ring <inline-formula><tex-math id="M9">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Some conditions for such linear codes over <inline-formula><tex-math id="M10">\begin{document}$ R $\end{document}</tex-math></inline-formula> to be linear complementary dual are given. Furthermore, based on the Artin conjecture, we get a class of good codes by calculating the total number of LCD double circulant codes over <inline-formula><tex-math id="M11">\begin{document}$ R $\end{document}</tex-math></inline-formula>.</p>

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.

Column cyclic rank metric codes and linear complementary dual rank metric codes

We introduce and study column cyclic rank metric (CCRM) codes over finite fields. We present natural questions for the existence and characterization of CCRM codes. We completely solve these questions only in the first nontrivial case: The case of the minimum rank distance [Formula: see text] and the number of rows [Formula: see text]. We also consider linear complementary dual rank metric (LCDRM) codes both in the setting of CCRM codes and in the general setting of rank metric codes. We also ask the natural questions for the existence and characterization of CCRM codes in the subclass of LCDRM codes. We also completely solve these questions for this subclass of LCDRM codes only in the first nontrivial case that [Formula: see text] and [Formula: see text]. Moreover we extend a characterization of linear complementary dual (LCD) codes of Massey to arbitrary rank metric codes in the general setting.

The Hull of Two Classical Propagation Rules and Their Applications

In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the (u, u + v)-construction and the direct sum construction. Some new criteria for the resulting codes derived from these two propagation rules being self-dual, self-orthogonal, or linear complementary dual (LCD) codes are given. As applications, we employ the (u, u + v)-construction to obtain (almost) self-orthogonal codes; employ the direct sum construction to provide lower bounds on the minimum distance of FSD (LCD) codes; and employ both these two constructions to derive linear codes with prescribed hull dimensions. Many (almost) optimal codes are presented. In particular, a family of binary almost Euclidean self-orthogonal Griesmer codes is constructed. We also obtain many binary, ternary Euclidean and quaternary Hermitian FSD LCD codes of larger lengths and improve some lower bounds on the minimum distance of known ternary Euclidean LCD codes.

On Locality of Some Binary LCD Codes

The design of codes for distributed storage systems that protects from node failures has been studied for years, and locally repairable code (LRC) is such a method that gives a solution for fast recovery of node failures. Linear complementary dual code (LCD code) is useful for preventing malicious attacks, which helps to secure the system. In this paper, we combine LRC and LCD code by integration of enhancing security and repair efficiency, and propose some techniques for constructing LCD codes with their localities determined. On the basis of these methods and inheriting previous achievements of optimal LCD codes, we give optimal or near-optimal [n, k, d, r] LCD codes for k ≤ 6 and n ≥ k+1 with relatively small locality, mostly r ≤ 3. Since all of our obtained codes are distance-optimal, in addition, we show that the majority of them are r-optimal and the other 63 codes are all near r-optimal, according to CM bound.

Theory of additive complementary dual codes, constructions and computations

We study complementary dual Fq-linear codes defined over an extension field Fqm, which we refer to as additive complementary dual (ACD) codes. We first consider duality of this class of codes with respect to a general inner product on Fqmn, which covers the commonly studied cases such as the trace-Euclidean and Hermitian inner products. In general this inner product is not necessarily symmetric, hence we define the notions of left/right hulls and left/right duals of an additive code. This leads to one-sided ACD codes, for which we prove a characterization result in terms of the generator matrix, extending the analogous characterization of Massey for linear complementary dual (LCD) codes. Then we focus on constructions and parameters of ACD codes with respect to trace-Euclidean, Hermitian and Galois inner products. We prove how LCD codes yield ACD codes relative to inner products considered in this work, make some observations on MDS subclass of ACD codes and present a construction of an ACD code by expanding another ACD code. We provide extensive computational results and tables on ACD codes over F4,F8 and F9. Our computations yield many MDS ACD codes and also examples of ACD codes, which have more codewords than the comparable LCD codes with the same length and minimum distance.

Connections between Linear Complementary Dual Codes, Permanents and Geometry

Linear codes with complementary duals, or LCD codes, have recently been applied to side-channel and fault injection attack-resistant cryptographic countermeasures. We explain that over characteristic two fields, they exist whenever the permanent of any generator matrix is non-zero. Alternatively, in the binary case, the matroid represented by the columns of the matrix has an odd number of bases. We explain how Grassmannian varieties as well as linear and quadratic complexes are connected with LCD codes. Accessing the classification of polarities, we relate the binary LCD codes of dimension k to the two kinds of symmetric non-singular binary matrices, to certain truncated Reed–Muller codes, and to the geometric codes of planes in finite projective space via the self-orthogonal codes of dimension k.

New entanglement-assisted quantum codes constructed from Hermitian LCD codes

<abstract><p>Hermitian linear complementary dual (LCD) codes are a class of linear codes that intersect with their Hermitian dual trivially. Each Hermitian LCD code can give an entanglement-assisted quantum error-correcting code (EAQECC) with maximal entanglement. Methods of constructing Hermitian LCD codes from known codes were developed, and seven new Hermitian LCD codes with parameters $ [119,4,88]_{4} $, $ [123,4,91]_{4} $, $ [124,4,92]_{4} $, $ [136,4,101]_{4} $, $ [140,4,104]_{4} $, $ [188,4,140]_{4} $ and $ [212,4,158]_{4} $ were constructed. Seven families of Hermitian LCD codes and their related EAQECCs were derived from these codes. These new EAQECCs have better parameters than those known in the literature.</p></abstract>

Galois LCD codes over mixed alphabets

In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of FpFp[θ]-linear codes, where p∈{2;3} and θ≠θ2=0 that provides LCD codes over Fp.

Additive Complementary Dual Codes From Group Characters

Additive codes have become an increasingly important topic in algebraic coding theory due to their applications in quantum error-correction and quantum computing. Linear Complementary Dual (LCD) codes play an important role for improving the security of information against certain attacks. Motivated by these facts, we define additive complementary dual codes (ACD for short) over a finite abelian group in terms of an arbitrary duality on the ambient space and examine their properties. We show that the best minimum weight of ACD codes is always greater than or equal to the best minimum weight of LCD codes of the same size and that this inequality is often strict. We give some matrix constructions for quaternary ACD codes from a given quaternary ACD code and also from a given binary self-orthogonal code. Moreover, we construct an algorithm to determine if a given quaternary additive code is an ACD code with respect to all possible symmetric dualities. We also determine the largest minimum distance of quaternary ACD codes for lengths <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n \leq 10$ </tex-math></inline-formula> . The obtained codes are either optimal or near optimal according to Bierbrauer <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> +. (2009).

LCD matrix product codes with an application

Matrix product (MP) codes over a finite field [Formula: see text], where [Formula: see text] a prime power, were first introduced by Blackmore and Norton [MP codes over [Formula: see text], Appl. Algebra Engrg. Comm. Comput. 12 (2001) 477–500]. This paper investigates linear complementary dual (LCD) MP codes over finite commutative Frobenius rings. We derive a necessary and sufficient condition for an MP code to have an LCD. We provide some efficient constructions of LCD MP codes. Finally, we have shown an application on cryptography of LCD codes.

An application of a virus optimization algorithm to the problem of computing binary self-dual and LCD codes

In this paper, we employ a virus optimization (VO) algorithm, which is one of the metaheuristic optimization techniques, and a known construction method to compute many new binary $ [72,36,12] $ self-dual codes and optimal/near-optimal linear complementary dual (LCD) codes. In particular, we obtain 39 Type Ⅰ and 19 Type Ⅱ codes of length 72, with parameters in their weight enumerators that were not known in the literature before, and 85 new binary LCD codes that are either optimal or near-optimal. We also present the generator matrix of extended Golay code $ [24,12,8] $ by a cyclic group matrix ring element.

On the existence of quaternary Hermitian LCD codes with Hermitian dual distance 1

For k≥2 and a positive integer d0, we show that if there exists no quaternary Hermitian linear complementary dual [n,k,d] code with d≥d0 and Hermitian dual distance greater than or equal to 2, then there exists no quaternary Hermitian linear complementary dual [n,k,d] code with d≥d0 and Hermitian dual distance 1. As a consequence, we generalize a result by Araya, Harada and Saito on the nonexistence of some quaternary Hermitian linear complementary dual codes.

Entanglement-Assisted Quantum Codes From Algebraic Geometry Codes

Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.

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

Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. We study the largest minimum weight d2(n,k) among all binary LCD [n,k] codes and the largest minimum weight d3(n,k) among all ternary LCD [n,k] codes. The largest minimum weights d2(n,5) and d3(n,4) are partially determined. We also determine the largest minimum weights d2(n,n−5), d3(n,n−i) for i∈{2,3,4}, and d3(n,k) for n∈{11,12,…,19}.

The concatenated structure of quasi-abelian codes

The decomposition of a quasi-abelian code into shorter linear codes over larger alphabets was given in Jitman and Ling (Des Codes Cryptogr 74:511–531, 2015), extending the analogous Chinese remainder decomposition of quasi-cyclic codes (Ling and Solé in IEEE Trans Inf Theory 47:2751–2760, 2001). We give a concatenated decomposition of quasi-abelian codes and show, as in the quasi-cyclic case, that the two decompositions are equivalent. The concatenated decomposition allows us to give a general minimum distance bound for quasi-abelian codes and to construct some optimal codes. Moreover, we show by examples that the minimum distance bound is sharp in some cases. In addition, examples of large strictly quasi-abelian codes of about a half rate are given. The concatenated structure also enables us to conclude that strictly quasi-abelian linear complementary dual codes over any finite field are asymptotically good.

Optimal binary LCD codes

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. These codes were first introduced by Massey in 1964. Nowadays, LCD codes are extensively studied in the literature and widely applied in data storage, cryptography, etc. In this paper, we prove some properties of binary LCD codes using their shortened and punctured codes. We also present some inequalities for the largest minimum weight $d_{LCD}(n,k)$ of binary LCD [n,k] codes for given length n and dimension k. Furthermore, we give two tables with the values of $d_{LCD}(n,k)$ for $k\le 32$ and $n\le 40$, and two tables with classification results.