2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75292The structure of multivariate semisimple 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 non-trivial codes over $R$. The structure of the dual codes in the semisimple abelian case is also derived and some conditions on the existence of selfdual codes over $R$ are studied.Submitted to SIAM Journal on Discrete MathematicsCombinatoricsRings and Algebras11T71, 13M10, 94B99text