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.

A finite semifield D is a finite nonassociative ring with identity such that the set $D^{\ast }=D\setminus \{0\}$ is closed under the product. In this paper we obtain a computer-assisted description of all semifields of order 64, which completes the classification of finite semifields of order at most 125.

Finite semifields (finite non-necessarily associative division rings) have traditionally been considered in the context of finite geometries (they coordinatize projective semifield planes). New applications to the coding theory, combinatorics and the graph theory have broadened the potential interest in these rings. We show recent progress in the study of these objects with the help of computational tools. In particular, we state results on the classification and primitivity of semifields obtained with the help of advanced and efficient implementations (both sequential and parallel) of different algorithms specially designed to manipulate these objects.