Self-orthogonal codes over a non-unital ring and combinatorial matrices

TL;DR

This paper explores self-orthogonal codes over a specific non-unital ring, utilizing combinatorial matrices from association schemes and graphs, leading to new code constructions with improved minimum distance bounds.

Contribution

It introduces a novel construction of self-orthogonal and Type IV codes over a non-unital ring using combinatorial matrices, enhancing bounds on minimum distance.

Findings

Constructed quasi self-dual and Type IV codes over ring E

Represented codes as formally self-dual additive codes over GF(4)

Improved classical bounds on minimum distance for Type IV codes

Abstract

There is a local ring $E$ of order $4,$ without identity for the multiplication, defined by generators and relations as $E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle.$ 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 $E.$