Construction of quasi self-dual codes over a commutative non-unital ring of order 4

Let I be the commutative non-unital ring of order 4 defined by generators and relations. I = < a, b vertical bar 2a = 2b = 0, a(2) = b, ab = 0 >. Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi-Type IV codes (QSD codes with even torsion code) over I up to lengths n = 6, and suggested two building-up methods for constructing QSD codes. In this paper, we construct more QSD codes, Type IV codes and quasi-Type IV codes for lengths n = 7 and 8, and describe five new variants of the two building-up construction methods. We find that when n = 8 there is at least one QSD code with minimun distance 4, which attains the highest minimum distance so far, and we give a generator matrix for the code. We also describe some QSD codes, Type IV codes and quasi-Type IV codes with new weight distributions.

Automated summary

This paper studies a commutative non-unital ring I and constructs QSD codes, Type IV codes and quasi-Type IV codes over I. The authors propose new construction methods and discover a QSD code with a minimum distance of 4, which achieves the highest minimum distance so far. They also describe codes with new weight distributions.