Completing the determination of the next-to-minimal weights of affine cartesian codes (Q1995210)

Let \(\mathbb{F}_q\) be the finite field with \(q\) elements, and \(K_i \subset \mathbb{F}_q\) with cardinality \(d_i \geq 2\) \((i=1,\ldots,n)\). We set \(\mathcal{X} = K_1\times \cdots \times K_n\), \(N = d_1\cdots d_n\), and we consider the polynomials \(f_i =\prod_{\alpha\in K_i} (X_i-\alpha)\) \((i=1,\ldots,n)\). The ideal of all polynomials in \(\mathbb{F}_q[X_1, \ldots , X_n]\) vanishing in \(\mathcal{X}\) is \(I_{\mathcal{X}} = (f_1, \ldots , f_n)\). Write \(\mathcal{X} = \{\alpha_1,\ldots,\alpha_N\}\) and consider the morphism \(\psi : \mathbb{F}_q[X_1, \ldots , X_n]/I_{\mathcal{X}}\) defined by \(\psi(f+I_{\mathcal{X}}) = f(\alpha_1,\ldots,\alpha_N)\). If \(d\) is a positive integer, then the affine cartesian code of order \(d\) is the image, by \(\psi\), of \(\mathbb{F}_q[X_1, \ldots , X_n]_{\leq d}\), and is denoted by \(C_{\mathcal{X}} (d)\). In [\textit{C. Carvalho} and \textit{V. G. L. Neumann}, Finite Fields Appl. 44, 113--134 (2017; Zbl 1354.14040)], the values of the next-to-minimal Hamming weights of affine Cartesian codes of order \(d\), for almost all values of \(d\) are determined. In the paper under review, the next-to-minimal weight of \(C_{\mathcal{X}} (d)\) for the remaining values of \(d\) are computed, under the hypotheses that \(3 \leq d_1 \leq \ldots \leq d_n\), that \(K_i\) \((i=1,\ldots,n)\) is a field, and that \(K_1 \subset \ldots \subset K_n \subset \mathbb{F}_q\).