On irreducible polynomial remainder codes

Abstract

A general class of polynomial remainder codes is considered. These codes are very flexible in rate and length and include Reed-Solomon codes as a special case. In general, the code symbols of such codes are polynomials of different degree, which leads to two different notions of weights and of distances. The notion of an error locator polynomial is generalized to such codes. A key equation is proposed, from which the error locator polynomial can be computed by means of a gcd algorithm. From the error locator polynomial, the transmitted message can be recovered in two different ways, which may be new even when specialized to Reed-Solomon codes.