The build-up construction of quasi self-dual codes over a non-unital ring

Abstract

There is a local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by generators and relations as [Formula: see text] We study a recursive construction of self-orthogonal codes over [Formula: see text] We classify, up to permutation equivalence, self-orthogonal codes of length [Formula: see text] and size [Formula: see text] (called here quasi self-dual codes or QSD) up to the length [Formula: see text]. In particular, we classify Type IV codes (QSD codes with even weights) up to [Formula: see text].

The build-up construction over a commutative non-unital ring

“…Starting from a self-dual code of length n, it builds a self-dual code of length n + 2 by a simple recursion. This method was used successfully over a non-unital non commutative ring in [1]. In this article, we adapt this method to generate quasi self-dual (QSD) codes over the ring I, a non-unital, commutative ring of order 4 [3].…”