Type IV codes over a non-unital ring

Abstract

There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: see text] and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

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

“…The following result went unnoticed in [1], and improves on the previously known upper bound d ≤ 2⌊ n+2 4 ⌋ for the minimum Hamming distance d of a Type IV E-code of length n.…”

“…The ring E is a non unital, non-commutative ring of order 4, of characteristic two [1,5]. Thus, E consists of four elements E = {0, a, b, c}, with c = a + b.…”

“…The first statement follows by specialization of variables in the MacWilliams relation for the joint weight enumerator of the residue and torsion code [1,Prop. 2].…”

“…In [3] this notion was extended over the three rings of order four that are not a field, namely Z 4 , F 2 + uF 2 , and F 2 + vF 2 . Recently, a further extension was accomplished over a non commutative non-unital ring in [1]. The concept of self-dual code is replaced there by quasi self-dual (QSD) code that is self-orthogonal of length n, with 2 n codewords.…”

“…This is a generalization from fields to rings of the approach of [4]. We construct QSD codes and Type IV codes over E. Along the way, we improve the upper bound on the minimum distance of Type IV codes from [1] by a multiplicative factor, by an application of the classical invariant bound for the minimum distance of extremal Type IV codes over F 4 . Some numerical results validate our approach.…”