The structure of multivariate serial codes over a finite chain ring R is established using the structure of the residue field $\bar R$. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some nontrivial codes over R. The structure of the dual codes in the serial abelian case is also derived, and some conditions for the existence of self‐dual codes over R are studied.

Codes over rings is still an emerging area in coding theory. Most of the work done so far has been done on alphabets that are chain rings. As the next logical step in the development of codes over rings, some studies have been done on codes over local Frobenius rings that are not chain rings. In this work, the smallest local Frobenius rings are shown to be of order 16 and a complete list of these seven rings is given. Along the way, all local rings of order 16 are found. In addition to these rings, some commonalities, pertaining to codes, between the local Frobenius non-chain rings are shown.

In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the (u, u + v)-construction and the direct sum construction. Some new criteria for the resulting codes derived from these two propagation rules being self-dual, self-orthogonal, or linear complementary dual (LCD) codes are given. As applications, we employ the (u, u + v)-construction to obtain (almost) self-orthogonal codes; employ the direct sum construction to provide lower bounds on the minimum distance of FSD (LCD) codes; and employ both these two constructions to derive linear codes with prescribed hull dimensions. Many (almost) optimal codes are presented. In particular, a family of binary almost Euclidean self-orthogonal Griesmer codes is constructed. We also obtain many binary, ternary Euclidean and quaternary Hermitian FSD LCD codes of larger lengths and improve some lower bounds on the minimum distance of known ternary Euclidean LCD codes.