Cyclic codes as invariant subspaces

Abstract

The description of the linear cyclic codes as ideals in the algebra ${\mathfrak{F}}_{\mathfrak{n}}\mathfrak{=}\mathfrak{F}\mathfrak{[}\mathfrak{x}\mathfrak{]}\mathfrak{/}\mathfrak{(}{\mathfrak{x}}^{\mathfrak{n}}\mathfrak{-}\mathfrak{1}\mathfrak{)}$, where $F$ is a finite field, is well known in the coding theory. The map cyclic shift is a linear operator in $F^n$. Our aim is to consider a new method of describing the cyclic codes as invariant subspaces of $F^n$ regarding this operator.