This paper investigates double circulant codes of length $2n$ over ${\mathbb{Z}}_{p^m}$ where $p$ is an odd prime, $n$ goes to infinity, and $m\ge 1$ is a fixed integer. Using random coding, we obtain families of asymptotically good Lee codes over ${\mathbb{Z}}_{p^m}$ in the case of small and large alphabets, and asymptotically good Euclidean codes over ${\mathbb{Z}}_{p^m}$ for small alphabets. We use Euclidean codes to construct spherical codes, and Lee codes to construct insertion/deletion codes, by a projection technique due to (Yaglom, 1958) for spherical codes, and to (Sok et al., 2018) for deletion codes.

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 $3$.

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 $I$.

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 four to generate quasi self-dual codes. In the present paper we introduce three such propagation rules to generate self-orthogonal, one-sided self-dual, and self-dual codes over a special non-unitary ring of order 9. As an application, we classify the three categories of codes in lengths at most $7,$ up to monomial equivalence. Mass formulas for the three classes of codes considered ensure that the classification is complete.