Ideal codes over separable ring extensions

TL;DR

This paper explores the algebraic structure of ideal codes over separable ring extensions, proving they are generated by idempotents and providing an efficient algorithm for their computation.

Contribution

It establishes that ideal codes are direct summands in non-commutative algebras and introduces a method to compute their generating idempotents using separability elements.

Findings

Ideal codes are direct summands as left ideals.

Ideal codes are generated by idempotent elements.

An efficient algorithm for computing idempotents is proposed.

Abstract

This paper investigates the application of the theoretical algebraic notion of a separable ring extension, in the realm of cyclic convolutional codes or, more generally, ideal codes. We work under very mild conditions, that cover all previously known as well as new non trivial examples. It is proved that ideal codes are direct summands as left ideals of the underlying non-commutative algebra, in analogy with cyclic block codes. This implies, in particular, that they are generated by an idempotent element. Hence, by using a suitable separability element, we design an efficient algorithm for computing one of such idempotents.