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Circulant matrices are an important tool widely used in coding theory andcryptography. A circulant matrix is a square matrix whose rows are the cyclicshifts of the first row. Such a matrix can be efficiently stored in memorybecause it is fully specified by its first row. The ring of $n \times n$circulant matrices can be identified with the quotient ring$\mathbb{F}[x]/(x^n-1)$. In consequence, the strong algebraic structure of thering $\mathbb{F}[x]/(x^n-1)$ can be used to study properties of the collectionof all $n\times n$ circulant matrices. The ring $\mathbb{F}[x]/(x^n-1)$ is aspecial case of a group algebra and elements of any finite dimensional groupalgebra can be represented with square matrices which are specified by a singlecolumn. In this paper we study this representation and prove that it is aninjective Hamming weight preserving homomorphism of $\mathbb{F}$-algebras andclassify it in the case where the underlying group is abelian.  Our work is motivated by the desire to generalize the BIKE cryptosystem (acontender in the NIST competition to get a new post-quantum standard forasymmetric cryptography). Group algebras can be used to design similarcryptosystems or, more generally, to construct low density or moderate densityparity-check matrices for linear codes.

Efficient Description of some Classes of Codes using Group Algebras