Generators and weights of polynomial codes

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[x1, . . . ,Xn]/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 double-struck F sign[x1, . . . ,xn]/(x1 a1(1-x1 b1), . . . ,xn an(1-xn bn)) where double-struck F sign is an arbitrary field.