On the next-to-minimal weight of affine Cartesian codes
This paper is interested in the value of the next-to-minimal weight of affine Cartesian codes. The next-to-minimal weight of an error-correcting code \(\mathcal{C}\)\, is the second lowest Hamming weight of the non-zero codewords of \(\mathcal{C}\). An affine Cartesian code \(\mathcal{C}_{\chi}(d)\)\, is the image of the evaluation map from polynomials of degree up to \(d\)\, of \(\mathbb{F}_q[X_1, \dots,X_n]\)\, on the points of a Cartesian set \(\chi=A_1\times \dots \times A_n \subseteq \mathbb{F}_q^n\). When \(A_i=\mathbb{F}_q,\,\, i=1,\dots,n\)\, we have the generalized Reed-Muller codes. The minimal weight of \(\mathcal{C}_{\chi}(d)\)\, was already known, see [\textit{H. H. López} et al., Des. Codes Cryptography 71, No. 1, 5--19 (2014; Zbl 1312.94118)], result quoted here as Theorem 1.2. The present paper deepens in the study of the next-to-minimal weight of those codes. The paper assumes that \(d_i=\sharp(A_i)\), \(3\leq d_1\leq \dots \leq d_n\), \(d\geq d_1 -1\), \(n\geq 3\). Section 2 finds, considering the restriction to some particular \(n\)-tuples of \(\chi\), an upper bound for the next-to-minimal weight of \(\mathcal{C}_{\chi}(d)\) (Theorem 2.4) and Section 3, using methods from Gröbner basis, determines a lower bound (Proposition 3.9). These two bounds allow then to know the exact value of the next-to-minimal weight of \(\mathcal{C}_{\chi}(d)\) codes in some particular cases (Theorem 3.10), generalizing similar results of \textit{R. Rolland} for generalized Reed-Muller codes [Cryptogr. Commun. 2, No. 1, 19--40 (2010; Zbl 1192.94107)].
scientific article; zbMATH DE number 6668667
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