Generators and weights of polynomial codes
Let \(f_1(x_1),\dots,f_n(x_n)\) be univariate polynomials over a field \({\mathbb{F}}\), and let \(R\) denote the Artinian ring \(\mathbb{F}[x_1,\dots,x_n]/(f_1(x_1),\dots,f_n(x_n))\). The authors show that the radical of \(R\) is a principal ideal if and only if at most one of the \(f_1,\dots,f_n\) is not square-free. In the case when \({\mathbb{F}}\) is a finite field, this result has applications to coding theory, since some interesting classes of codes, including generalized Reed-Muller codes [cf. \textit{P. Charpin}, Commun. Algebra 16, 2231-2246 (1988; Zbl 0649.94013)], occur as powers of the radical in such an Artinian ring. The authors also consider the case when the field \(\mathbb{F}\) is replaced by the ring \(\mathbb{Z}_m\), where \(m\) is a power of a prime. Finally, using results of \textit{H. N. Ward} [Arch. Math. 54, 307-312 (1990; Zbl 0682.94012)], the authors compute the Hamming weight of the radical and its powers in the ring \[ \mathbb{F}[x_1,\dots,x_n]/(x_1^{a_1}(1-x_1^{b_1}),\dots, x_n^{a_n}(1-x_n^{b_n})). \]
scientific article; zbMATH DE number 1145415