Modified Genetic Algorithms for Constructing New Quaternary Hermitian LCD Codes

Scalar products and Left LCD codes

In this paper, we shall give an explicit proof that constacyclic codes over finite commutative rings can be realized as ideals in some twisted group rings. Also, we shall study isometries between those codes and, finally, we shall study k-Galois LCD constacyclic codes over finite fields. In particular, we shall characterize constacyclic LCD codes with respect to Euclidean inner product in terms of its idempotent generators and the classical involution using the twisted group algebras structures and find some good LCD codes

A Study of Galois LCD Codes Over a Family of Non-chain Rings

Abstract A multisecret-sharing scheme is a method for distributing n randomly associated secrets s 1, s 2,..., s n among a set of attendants. Linear codes with complementary duals, LCD codes, form an important class of linear codes, while weighing orthogonal matrices play a significant role in cryptography. In this paper, we design novel multisecret-sharing schemes based on LCD codes derived from weighing orthogonal matrices. We analyze the access structures, coalition statistics, security aspects, and information-theoretic efficiency compared to existing multisecret-sharing methods.

On the classification of binary LCD codes

Asymptotically good four Toeplitz codes and derived new formally self-dual LCD codes

Characterizations of the Minimum Weights of LCD Codes of Large Dimensions

In this paper, we study self-dual and LCD codes constructed from Kneser graphs K(n, 2) and collinearity graphs of generalized quadrangles using the so-called pure and bordered construction. We determine conditions under which these codes are self-dual or LCD. Further, for the codes over Z2k, we give the conditions under which they are Type II. Moreover, we study binary and ternary self-dual and LCD codes from Kneser graphs K(n, 2) and collinearity graphs of generalized quadrangles. Furthermore, from the support designs for certain weights of some of the codes, we construct strongly regular graphs and 3-designs.

On the minimum weights of quaternary Hermitian LCD codes

Two classes of LCD BCH codes over finite fields

Self-orthogonal codes have nice applications in many areas including quantum codes, lattices and LCD codes. For a prime power <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> , it is in general difficult to construct <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary self-orthogonal codes. In the literature, there exists no simple method to judge whether a general <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary linear code is self-orthogonal or not. In this paper, we mainly present several families of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary self-orthogonal codes and study their applications in quantum codes and LCD codes. Firstly, several families of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary linear codes are constructed by some special defining sets. These codes are proved to be self-orthogonal. To this end, we determine the numbers of solutions of some systems of equations over finite fields. Secondly, three families of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary quantum codes with unbounded length and minimum distance three are constructed from the self-orthogonal codes. These quantum codes are optimal according to the quantum Hamming bound. In particular, some of them have better parameters than known ones. Thirdly, several families of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary LCD codes are constructed from the self-orthogonal codes. Many optimal or almost optimal binary and ternary LCD codes are produced by our constructions. Some binary and ternary LCD codes have better parameters than known ones.

In this paper, we consider the hull of an algebraic geometry code, meaning the intersection of the code and its dual. We demonstrate how codes whose hulls are algebraic geometry codes may be defined using only rational places of Kummer extensions (and Hermitian function fields in particular). Our primary tool is explicitly constructing non-special divisors of degrees g and g − 1 on certain families of function fields with many rational places, accomplished by appealing to Weierstrass semigroups. We provide explicit algebraic geometry codes with hulls of specified dimensions, producing along the way linearly complementary dual algebraic geometric codes from the Hermitian function field (among others) using only rational places and an answer to an open question posed by Ballet and Le Brigand for particular function fields. These results complement earlier work by Mesnager, Tang, and Qi that use lower-genus function fields as well as instances using places of a higher degree from Hermitian function fields to construct linearly complementary dual (LCD) codes and that of Carlet, Mesnager, Tang, Qi, and Pellikaan to provide explicit algebraic geometry codes with the LCD property rather than obtaining codes via monomial equivalences.

Binary and ternary leading-systematic LCD codes from special functions

New constructions of optimal binary LCD codes

We show that a necessary and sufficient condition for a cyclic code C of length N over a finite chain ring R (whose maximal ideal has nilpotence 2) to be an LCD code is that C = (f(x)), where f(X) is a self-reciprocal monic divisor of XN − 1 in R[X] and x = X + (XN − 1) in R[X]/(XN − 1). A similar, but slightly different, theorem was proved in 2019 by Z. Liu and J. Wang for general finite chain rings (Theorem 25 in [5]). We provide two proofs, both completely different than the proof of Liu and Wang.

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.

Ternary self-orthogonal codes from weakly regular bent functions and their application in LCD Codes

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.

Let Fq be a field of order q, where q is a power of an odd prime p, and α and β are two non-zero elements of Fq. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R=Fq[u1,u2]/⟨u12−α2,u22−β2,u1u2−u2u1⟩. We decompose the ring R by using orthogonal idempotents Δ1,Δ2,Δ3, and Δ4 as R=Δ1R⊕Δ2R⊕Δ3R⊕Δ4R, and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes.