Generators and weights of polynomial codes

摘要

Berman and Charpin proved that all generalized Reed-Muller codes coincide with powers of the radical of a certain algebra. The ring-theoretic approach was developed by several authors including Landrock and Manz, and helped to improve parameters of the codes. It is important to know when the codes have a single generator. We consider a class of ideals in polynomial rings containing all generalized Reed-Muller codes, and give conditions necessary and sufficient for the ideal to have a single generator. The main result due to Glastad and Hopkins (Comment. Math. Univ. Carolin. 21, 371 - 377 (1980)) is an immediate corollary to our theorem.We also describe all finite quotient rings Z/mZ[x_1,...,x_n]/I which are commutative principal ideal rings where I is an ideal generated by univariate polynomials and then give formulas for the minimum Hamming weight of the radical and its powers in the algebra F[x_1,...,x_n]/(x_1a_1(1-x_1b_1), x_na_n(1-x_nb_n)) where F is an arbitrary field.