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.

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, a natural question to ask is which optimal linear codes have an efficient decoding. We show that two binary optimal $[36,19,8]$ linear codes and two binary optimal $[40,22,8]$ codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal $[36,19,8]$ and $[40,22,8]$ codes. We project them onto the much shorter length linear $[9,5,4]$ and $[10,6,4]$ codes over $GF\left(4\right)$ , respectively. This decoding algorithm, called projection decoding, can correct errors of weight up to 3. These $[36,19,8]$ and $[40,22,8]$ codes respectively have more codewords than any optimal self-dual $[36,18,8]$ and $[40,20,8]$ codes for given length and minimum weight, implying that these codes are more practical.