Quasi self-dual codes over non-unital rings from three-class association schemes.

Let E and I denote the two non-unital rings of order 4 in the notation of (Fine, 93) defined by generators and relations as E = (a, b | 2a = 2b = 0, a² = a, b² = b, ab = a, ba = b) and I = (a, b | 2a = 2b = 0, a2 = b, ab = 0). Recently, Alahmadi et al classified quasi self-dual (QSD) codes over the rings E and I for lengths up to 12 and 6, respectively. The codes had minimum distance at most 2 in the case of I, and 4 in the case of E. In this paper, we present two methods for constructing linear codes over these two rings using the adjacency matrices of three-class association schemes. We show that under certain conditions the constructions yield QSD or Type IV codes. Many codes with minimum distance exceeding 4 are presented. The form of the generator matrices of the codes with these constructions prompted some new results on free codes over E and I. [ABSTRACT FROM AUTHOR]