Narrow Results

There is a local ring $E$ of order $4,$ without identity for the multiplication, defined by generators and relations as

There is a non-unital ring $I$ of order 4 defined by generators and relations as $I=〈a,b|2a=2b=0,a^2=b,ab=0〉.$ In this paper, we present special constructions of linear codes over $I$ from the adjacency matrices of two class association schemes. These consist of either Strongly Regular Graphs (SRGs) or Doubly Regular Tournaments (DRTs). We investigate the conditions under which these codes are self-orthogonal, quasi self-dual, or Type IV. As a byproduct of this study, some diophantine relations on the weight of sum of vectors with coefficients in $I$ are derived. Some examples of codes with minimum distance better than that of Type IV codes over unital rings of the same order in modest lengths are given.

In the last 60 years coding theory has been studied a lot over finite fields ${𝔽}_q$ or commutative rings ${ℛ}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring with unity) of order $p^2$ had been presented, the construction of codes over non-commutative rings or non-commutative non-unital rings surfaced merely two years ago. In this letter, we extend the diverse research on exploring the codes over the non-commutative and non-unital ring $E=〈2a=2b=0,a^2=a,b^2=b,ab=a,ba=b〉$ by presenting the classification of optimal and nice codes of length $n\le 7$ over $E$, along with respective weight enumerators and complete weight enumerators.