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 $E=⟨a,b\mid 2a=2b=0,a^2=a,b^2=b,ab=a,ba=b⟩$ and $I=⟨a,b\mid 2a=2b=0,a^2=b,ab=0⟩$. 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 $I$.