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

Abstract

There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as $$\begin{aligned} I=\langle a,b \mid 2a=2b=0,\, a^2=b,\, \,ab=0 \rangle . \end{aligned}$$We study a recursive construction of self-orthogonal codes over I. We classify self orthogonal codes of length n and size \(2^n\) (called here quasi self-dual codes or QSD) up to the length \(n=6.\) In particular, we classify Type IV codes (QSD codes with even weights) and quasi Type IV codes (QSD codes with even torsion code) up to \(n=6.\)