The mass formula for self-orthogonal and self-dual codes over a non-unitary commutative ring

In this paper, we establish a mass formula for self-orthogonal codes, quasi self-dual codes, and self-dual codes over commutative non-unital rings ${\mathit{I}}_p=⟨a,b|pa=pb=0,a^2=b,ab=0⟩$, where $p$ is an odd prime. We also give a classification of the three said classes of codes over ${\mathit{I}}_p$ where $p=3,5,$ and $7$, with lengths up to $3$.