Classical codes as ideals in group algebras (Q1200298)

Let \(p\) be a prime and let \({\mathbf F}_ q\) be a finite field with \(q=p^ m\) elements. The group ring \({\mathbf F}_ p[{\mathbf F}_ q]\) is isomorphic to the knotted polynomial ring \({\mathbf F}_ p[X_ 1,\dots,X_ m]/(X^ p_ 1,\dots,X^ p_ m)\). This is a local ring with maximal ideal \(M=(X_ 1,\dots,X_ m)\). Using only some elementary properties of this ring, the authors show that the ideals \(M^ j\) are naturally isomorphic to the \(((p-1)m-j)\)-th order Generalized Reed-Muller codes over \({\mathbf F}_ p\). They prove a somewhat weaker result when the basefield is a proper extension of \({\mathbf F}_ p\). As an application of their approach, the authors present a new decoding algorithm for binary Reed-Muller codes.