On semisimple algebra codes: generator theory

Abstract

The class of affine variety codes is defined as the \(\mathbb F_q\) linear subspaces of \(\mathcal A\) a \(\mathbb F_q\)-semisimple algebra, where \(\mathbb F_q\) is the finite field with \(q=p^r\) elements and characteristic \(p\). It seems natural to impose to the code some extra structure such as been a subalgebra of \(\mathcal A\). In this case we will have codes that have a Mattson-Solomon transform treatment as the classical cyclic codes. Moreover, the results on the structure of semisimple finite dimensional algebras allow us to study those codes from the generator point of view.