How to expand a self-orthogonal code

Combinatorial designs have been widely used, in the construction of self-dual codes. Recently, new methods of constructing self-dual codes are established using orthogonal designs (ODs), generalized orthogonal designs (GODs), a set of four sequences and Diophantine equations over GF(p). These methods had led to the construction of many new self-dual codes over small finite fields and rings. In this paper, we used some methods to construct self-orthogonal and self dual codes over GF(p), for some primes p. The construction is achieved by using some special kinds of combinatorial designs like orthogonal designs and GODs. Moreover, we combine eight circulant matrices, a system of Diophantine equations over GF(p), and a recently discovered array to obtain a new construction method. Using this method new self-dual and self-orthogonal codes are obtained. Specifically, we obtain new self-dual codes l32,16,12r over GF(11) and GF(13) which improve the previously known distances.

Self-orthogonal codes form an important class of linear codes due to their rich algebraic structures and wide applications. In this paper, the well-known matrix-product construction for linear codes is applied to construct self-orthogonal codes. Necessary conditions on input codes and matrices for constructing self-orthogonal matrix-product codes are given as well as some illustrative examples.

Self-orthogonal codes over [formula omitted] arising from the chain ring [formula omitted

Self-dual codes over GF(5)

A heuristic algorithm is designed to searching for good higher dimensional self-orthogonal codes over GF(A) from low dimensional self-orthogonal codes. Many self-orthogonal codes of length 20 les n les 36 and dual distance 5 or 6 are obtained, and several have improved dual distance. Consequently, using these self-orthogonal codes and their dual codes, some linear quantum codes of minimum distance five or six for such length n are obtained, and three of these codes [[24, 8, 5]], [[25, 9, 5]], [[27, 7, 6]] are record-breaking. Quaternary code, self-orthogonal code, linear quantum error-correcting code.

In this paper we generalize the construction of binary self-orthogonal codes obtained from weakly self-orthogonal designs described in Tonchev (J Combinat Theory Ser A 52:197-205, 1989) in order to obtain self-orthogonal codes over an arbitrary field. We extend construction self-orthogonal codes from orbit matrices of self-orthogonal designs and weakly self-orthogonal 1-designs such that block size is odd and block intersection numbers are even described in Crnković (Adv Math Commun 12:607–628, 2018). Also, we generalize mentioned construction in order to obtain self-orthogonal codes over an arbitrary field. We construct weakly self-orthogonal designs invariant under an action of Mathieu group \(M_{11}\) and, from them, binary self-orthogonal codes.

Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}_{(r,r^{m}-1,\delta,b)}$ </tex-math></inline-formula> be the primitive BCH code over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{r}$ </tex-math></inline-formula> of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r^{m}-1$ </tex-math></inline-formula> with designed distance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{r}$ </tex-math></inline-formula> is the finite field of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> . In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}_{(r,r^{2}-1,\delta,1)}$ </tex-math></inline-formula> and its hull in both Hermitian and Euclidean cases for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \le \delta \le r^{2}-1$ </tex-math></inline-formula> . We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.

Multithreshold decoders (MTD) for the self-orthogonal codes (SOC) are considered. It is shown that applying of MTD for the well chosen codes provides almost optimum decoding. Noted that MTD using is difficult to provide a low decoding error probability at the a large noise level due to a small code distance of SOC. For reducing the error probability we propose a concatenated coding scheme formed by SOC and a nonredundant accumulate code. Simulation results show that these concatenated codes allow decrease BER decoding on four decimal orders at the almost optimum decoding area of the used SOC.

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.

A lower bound on the length of binary self-orthogonal unequal error protection (UEP) codes is derived, and two design procedures for constructing optimal self-orthogonal UEP codes are proposed. With this lower bound, known self-orthogonal UEP codes can be evaluated. It is pointed out that, for given values of minimum distance and code rate, the self-orthogonal codes must be relatively long, so optimal self-orthogonal codes are not optimal in general. But self-orthogonal codes can be implemented simply, and they have error-correcting capabilities beyond those guaranteed by their minimum distance. These properties can be viewed as a partial compensation for using self-orthogonal codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary k-dimensional linear code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ( k = 3,4) into a self-orthogonal code of the shortest length which has the same dimension k and minimum distance d' ≥ d( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ). For k > 4, we suggest a recursive method to embed a k-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length n with dimension 4 and any length n \not ≡ 6, 13,14,21,22,28,29 (mod 31) with dimension 5.

In this paper, a special subclass of binary generalized quasi-cyclic self-orthogonal codes and quantum codes constructed by Steane construction are discussed. Firstly, eight 16-dimensional even length generalized quasi-cyclic self-orthogonal codes with dual distance five are built based on circulant or partial circulant matrices. Secondly, pairs of nested self-orthogonal codes with dual distance five and three are designed by applying an algorithm for searching subcodes of a given code. Thirdly, revised pairs of codes with dual distance six and four are constructed by extending previous pairs of codes, and then eight quantum codes with distance six are obtained by Steane construction. These eight quantum codes are new binary construction by Steane construction and are best known ones. Index Terms - Quantum code, Steane construction, self- orthogonal code, generalized quasi-cyclic code, pair of nested self- orthogonal codes.

Due to some practical applications, linear complementary dual (LCD) codes and self-orthogonal codes have attracted wide attention in recent years. In this paper, we use simplicial complexes for construction of an infinite family of binary LCD codes and two infinite families of binary self-orthogonal codes. Moreover, we explicitly determine the weight distributions of these codes. We obtain binary LCD codes which have minimum weights two or three, and we also find some self-orthogonal codes meeting the Griesmer bound. As examples, we also present some (almost) {\it optimal} binary self-orthogonal codes and LCD {\it distance optimal} codes.

The main objective of this article is to study self-orthogonal negacycliccodes of length $n$ over a finite field $\mathbb{F}_{q}$, wherethe characteristic of $\mathbb{F}_{q}$ does not divide $n$. We investigateissues related to their existence, characterization and enumeration.We find the necessary and sufficient conditions for the existenceof self-orthogonal negacyclic codes of length $n$ over a finite field$\mathbb{F}_{q}$. We characterize the defining sets and the correspondinggenerator polynomials of these codes. We obtain formulae to calculatethe number of self-dual and self-orthogonal negacyclic codes of agiven length $n$ over $\mathbb{F}_{q}$. The enumeration formulafor self-orthogonal negacyclic codes involves a two-variable function$\chi(d,q)$ defined by $\chi(d,q)=0$ if $d$ divides $(q^{k}+1)$for some $k\geq0$ and $\chi(d,q)=1$, otherwise. We give necessaryand sufficient conditions when $\chi(d,q)=0$ holds.

Fuzzy linear codes based on nested linear codes