Menichetti's nonassociative $G$-crossed product algebras and Menichetti codes

Abstract

We demonstrate the use of nonassociative algebras in code design and consider codes with nonassociative ambient algebras other than the well-known skew polycyclic codes. We define and investigate Menichetti algebras and identify them as important elements in the semiassociative Brauer monoid. Menichetti algebras can be viewed as generalisations of $G$-crossed product algebras; they are $n^2$-dimensional algebras with an $n$-dimensional Galois field extension $K/F$ with Galois group $G$ in their nucleus. We then extend the class of linear error-correcting codes obtained from left principal ideals in their ambient algebra using the opposite algebras of Menichetti algebras as ambient algebra. With the right choice of algebra they display symmetric and cyclic properties which promise efficient decoding algorithms. Well-known examples of such Menichetti codes are those skew constacyclic codes which have a nonassociative $G$-crossed product algebra (a nonassociative cyclic algebra) as their ambient algebra.