Abstract
Circulant matrices are an important tool widely used in coding theory and cryptography. A circulant matrix is a square matrix whose rows are the cyclic shifts of the first row. Such a matrix can be efficiently stored in memory because it is fully specified by its first row. The ring of n x n circulant matrices can be identified with the quotient ring F[x]/(x(n) - 1). In consequence, the strong algebraic structure of the ring F[x]/(x(n) - 1) can be used to study properties of the collection of all n x n circulant matrices. The ring F[x]/(x(n) - 1) is a special case of a group algebra and elements of any finite dimensional group algebra can be represented with square matrices which are specified by a single column. In this paper we study this representation and prove that it is an injective Hamming weight preserving homomorphism of F-algebras and classify it in the case where the underlying group is abelian.
Chimal-Dzul, Henry