Completing the determination of the next-to-minimal weights of affine cartesian codes
type: Item
instance of: scholarly article
title: Completing the determination of the next-to-minimal weights of affine cartesian codes
zbMATH Open document ID: 1458.14039
author: Victor G. L. Neumann
author: Cicero Carvalho
published in: Finite Fields and their Applications
publication date: 2021-02-19T00:00:00Z
review text: 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 \(K_1 \subset \ldots \subset K_n \subset \mathbb{F}_q\).
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 94B60
zbMATH DE Number: 7313138
zbMATH Keywords: affine cartesian codes
zbMATH Keywords: next-to-minimal weights
zbMATH Keywords: higher Hamming weights
zbMATH Keywords: Reed-Muller type codes
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
full work available at URL: j.ffa.2020.101775
OpenAlex ID: W3096243233
cites work: Q3000909
cites work: On the second Hamming weight of some Reed-Muller type codes
cites work: On the next-to-minimal weight of affine Cartesian codes
cites work: An Extension of Delsarte, Goethals and Mac Williams Theorem on Minimal Weight Codewords to a Class of Reed-Muller Type Codes
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DOI: 10.1016/J.FFA.2020.101775
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description: scientific article; zbMATH DE number 7313138