We study the structure of self-orthogonal and self-dual codes over two non-unital rings of order four, namely, the commutative ring $I=⟨a,b|2a=2b=0,a^2=b,\mathit{ab}=0⟩$ and the noncommutative ring $E=⟨a,b|2a=2b=0,a^2=a,b^2=b,\mathit{ab}=a,\mathit{ba}=b⟩.$ We use these structures to give mass formulas for self-orthogonal and self-dual codes over these two rings, that is, we give the formulas for the number of inequivalent self-orthogonal and self-dual codes, of a given type, over the said rings. Finally, using the mass formulas, we classify self-orthogonal and self-dual codes over each ring, for small lengths and types.

In this paper, we derive a mass formula for the self-orthogonal codes and self-dual codes over a non-commutative non-unitary ring, namely, $E_p=⟨a,b|pa=pb=0,a^2=a,b^2=b,ab=a,ba=b⟩,$ where $a\ne b$ and p is any odd prime. We also give a classification of self-orthogonal codes and self-dual codes over $E_p$, where $p=3,5,$ and 7, in short lengths.

The build-up method is a powerful class of propagation rules that generate self-dual codes over finite fields and unitary rings. Recently, it was extended to non-unitary rings of order 4, to generate quasi self-dual codes. In the present paper, we introduce three such propagation rules to generate self-orthogonal, self-dual and quasi self-dual codes over a special non-unitary ring of order 9. As an application, we classify the three categories of codes completely in length at most 3, and partially in lengths 4 and 5, up to monomial equivalence.

In this paper, we establish a mass formula for Euclidean and Hermitian self-orthogonal codes over the finite ring Fq + uFq, where Fq is the finite field of order q and u2 = 0. We also establish a mass formula for Euclidean self-orthogonal codes over the finite ring Fq + uFq + u2Fq, with u3 = 0 and characteristic of Fq is odd. These mass formulas are used to give a classification of Euclidean and Hermitian self-orthogonal codes over F2 + uF2 and F3 + uF3 of small lengths.