On quasi-abelian codes over finite commutative chain rings and their dual codes

Quasi-abelian codes are generalizations of several important families of error-correcting codes, including cyclic codes, quasi-cyclic codes, and abelian codes. In this paper, we study quasi-abelian codes over finite commutative chain rings and their dual codes by placing the Euclidean bilinear form on abelian group algebras over finite commutative chain rings. We further study and characterize all Euclidean self-orthogonal, self-dual, and LCD quasi-abelian codes over finite commutative chain rings. With the help of this characterization, we derive a necessary and sufficient condition for the existence of a Euclidean self-dual quasi-abelian code over a finite commutative chain ring of odd characteristic. We further count all Euclidean self-orthogonal and self-dual quasi-abelian codes over finite commutative chain rings of odd characteristic. We also obtain an enumeration formula for all Euclidean LCD quasi-abelian codes over finite commutative chain rings. We illustrate these results with some examples.