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Register a new userSelf-orthogonal codes over a non-unital ring and combinatorial matrices
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Added by
Minjia Shi
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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There is a local ring $E$ of order $4,$ without identity for the multiplication, defined by generators and relations as $E=langle a,b mid 2a=2b=0,, a^2=a,, b^2=b,,ab=a,, ba=brangle.$ We study a special construction of self-orthogonal codes over $E,$ based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over $E,$ and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over $F_4.$ The classical invariant theory bound for the weight enumerators of this class of codesimproves the known bound on the minimum distance of Type IV codes over $E.$
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We introduce a consistent and efficient method to construct self-dual codes over $GF(q)$ with symmetric generator matrices from a self-dual code over $GF(q)$ of smaller length where $q equiv 1 pmod 4$. Using this method, we improve the best-known minimum weights of self-dual codes, which have not significantly improved for almost two decades. We focus on a class of self-dual codes, including double circulant codes. Using our method, called a `symmetric building-up construction, we obtain many new self-dual codes over $GF(13)$ and $GF(17)$ and improve the bounds of best-known minimum weights of self-dual codes of lengths up to 40. Besides, we compute the minimum weights of quadratic residue codes that were not known before. These are: a [20,10,10] QR self-dual code over $GF(23)$, two [24,12,12] QR self-dual codes over $GF(29)$ and $GF(41)$, and a [32,12,14] QR self-dual codes over $GF(19)$. They have the highest minimum weights so far.
Let $mathbb{F}_q$ be a finite field of order $q$, a prime power integer such that $q=et+1$ where $tgeq 1,egeq 2$ are integers. In this paper, we study cyclic codes of length $n$ over a non-chain ring $R_{e,q}=mathbb{F}_q[u]/langle u^e-1rangle$. We define a Gray map $varphi$ and obtain many { maximum-distance-separable} (MDS) and optimal $mathbb{F}_q$-linear codes from the Gray images of cyclic codes. Under certain conditions we determine { linear complementary dual} (LCD) codes of length $n$ when $gcd(n,q) eq 1$ and $gcd(n,q)= 1$, respectively. It is proved that { a} cyclic code $mathcal{C}$ of length $n$ is an LCD code if and only if its Gray image $varphi(mathcal{C})$ is an LCD code of length $4n$ over $mathbb{F}_q$. Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over $R_{e,q}$.
In this paper, a criterion of MDS Euclidean self-orthogonal codes is presented. New MDS Euclidean self-dual codes and self-orthogonal codes are constructed via this criterion. In particular, among our constructions, for large square $q$, about $frac{1}{8}cdot q$ new MDS Euclidean (almost) self-dual codes over $F_q$ can be produced. Moreover, we can construct about $frac{1}{4}cdot q$ new MDS Euclidean self-orthogonal codes with different even lengths $n$ with dimension $frac{n}{2}-1$.
The parameters of MDS self-dual codes are completely determined by the code length. In this paper, we utilize generalized Reed-Solomon (GRS) codes and extended GRS codes to construct MDS self-dual (self-orthogonal) codes and MDS almost self-dual codes over. The main idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are Euclidean self-dual (self-orthogonal). The evaluation sets are consists of two subsets which satisfy some certain conditions and the length of these codes can be expressed as a linear combination of two factors of q-1. Four families of MDS self-dual codes, two families of MDS self-orthogonal codes and two families of MDS almost self-dual codes are obtained and they have new parameters.
Galois images of polycyclic codes over a finite chain ring $S$ and their annihilator dual are investigated. The case when a polycyclic codes is Galois-disjoint over the ring $S,$ is characterized and, the trace codes and restrictions of free polycyclic codes over $S$ are also determined givind an analogue of Delsarte theorem among trace map, any S -linear code and its annihilator dual.
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