In this paper, the determinants of n × n matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of n × n matrices over a commutative finite chain ring R of a fixed determinant a is determined for all a ∈ R and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of n × n matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.
The main focus of this paper is the complete enumeration of self-dual abelian codes in non-principal ideal group algebras F 2 k [A × Z 2 × Z 2 s ] with respect to both the Euclidean and Hermitian inner products, where k and s are positive integers and A is an abelian group of odd order. Based on the well-known characterization of Euclidean and Hermitian self-dual abelian codes, we show that such enumeration can be obtained in terms of a suitable product of the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length 2 s over some Galois extensions of the ring F 2 k + uF 2 k , where u 2 = 0. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of length p s over F p k + uF p k are given. Combining these results, the complete enumeration of self-dual abelian codes in F 2 k [A × Z 2 × Z 2 s ] is therefore obtained.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.