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

TL;DR

This paper explores the use of nonassociative algebras, specifically Menichetti algebras, in designing error-correcting codes with symmetric and cyclic properties, extending the class of codes beyond traditional associative frameworks.

Contribution

It introduces Menichetti algebras as a generalization of $G$-crossed product algebras and demonstrates their application in constructing novel error-correcting codes.

Findings

Menichetti algebras are identified as key elements in the semiassociative Brauer monoid.

Codes derived from these algebras exhibit symmetry and cyclicity, enabling efficient decoding.

Extension of linear codes using opposite algebras of Menichetti algebras broadens coding options.

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.