Generators and weights of polynomial codes

Abstract

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 $ \Bbb Z/m\Bbb Z[x_1,\dots,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 $\Bbb F[x_1,\ldots ,x_n]/(x_1^{a_1}(1-x_1^{b_1}), \ldots ,x_n^{a_n}(1-x_n^{b_n}))$ where $\Bbb F$ is an arbitrary field.