Polynomial weights and code constructions.

Abstract

Study of certain polynomials with the 'weight-retaining' property that any linear combination of these polynomials with coefficients in a general finite field has Hamming weight at least as great as that of the minimum-degree polynomial included. This fundamental property is used in applications to Reed-Muller codes, a new class of 'repeated-root' binary cyclic codes, two new classes of binary convolutional codes derived from binary cyclic codes, and two new classes of binary convolutional codes derived from Reed-Solomon codes.