Ideal Codes Over Separable Ring Extensions

Abstract

In this paper, an application of the theoretical algebraic notion of a separable ring extension in the realm of cyclic convolutional codes or, more generally, ideal codes, is investigated. It is worked 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 a non-commutative idempotent polynomial. Hence, by using a suitable separability element, an efficient algorithm for computing one of such idempotents is designed. We show that such an idempotent generator polynomial can be used to get information on the free distance of the convolutional code.