Fibre products of Artin-Schreier curves and generalized Hamming weights of codes
type: Item
instance of: scholarly article
title: Fibre products of Artin-Schreier curves and generalized Hamming weights of codes
zbMATH Open document ID: 0824.94025
DOI: 10.1016/0097-3165(95)90099-3
author: Gerard van der Geer
author: Marcel van der Vlugt
published in: Journal of Combinatorial Theory. Series A
publication date: 1995-11-15T00:00:00Z
review text: Given an \([n, k, d]\)-code \(C\) its \(r\)-th generalized Hamming weight \(d_ r (C)\) is defined as the minimum of \(w(D)= \# S(D)\), where \(D\) runs through the \(r\)-dimensional subcodes of \(C\) and \(S(D)\) denotes its support. The \(r\)-dimensional weight distribution \(A^{(r)}_ i\) of \(C\) is the number of subcodes \(D\) with \(\dim D=r\) and \(w(D)=i\). The main purpose of the authors is to compute these numbers for certain trace codes obtained from fibre product of Artin-Schreier curves in terms of the number of rational points of these curves. More precisely, let \(P\) be a subset of \(\mathbb P^ 1 (\mathbb F_ q)\) and \({\mathcal L}\) a finite dimensional \(\mathbb F_ q\)-linear subspace of the field of rational functions \(\mathbb F_ q (x)\) with \({\mathcal L}\cap \mathbb F_ q= \{0\}\). Suppose that the non-zero elements of \({\mathcal L}\) have poles in \(P\) only and with multiplicities prime to \(p\), where \(q\) is a power of \(p\). Consider the trace code \[ \text{Tr}(C)= \left\{c_ f= \left(\text{Tr}(f(x))_{x\in \mathbb P^ 1 \left(\mathbb F_ q\right)-P}\right), \;f\in {\mathcal L}\right\}. \] For each \(r\)-dimensional \(\mathbb F_ q\)-subspace \(D\) of \(\text{Tr}(C)\), one chooses a basis \(c_{f_ 1}, \ldots, c_{f_ r}\) corresponding to \(f_ 1, \ldots, f_ r\) of \({\mathcal L}\) and denote by \({\mathcal L}_ D\) the subspace spanned by the \(f_ i\)'s. To each \(f_ i\) one attaches an Artin-Schreier curve \(C_{f_ i}\) with affine equation \(y^ p- y= f_ i (x)\) and considers the maps \(\varphi_ i: C_{f_ i}\to \mathbb P^ 1\). The authors compute the numbers \(d_ r (\text{Tr} (C))\) in terms of the number of rational points of the curve \(C^{(D)}\), which is the normalization of \(C_{f_ 1} \times_{\mathbb P^ 1} \cdots \times_{\mathbb P^ 1} C_{f_ i}\). This curve defines a family of curves \(C_ f\) in the Grassmann variety \(G(r,n)\) of \(r\)-planes in \(n\)-space, where \(n\) is the length of \(\text{Tr}(C)\). Further they estimate the difference \(…
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 753624
zbMATH Keywords: generalized Hamming weight
zbMATH Keywords: weight distribution
zbMATH Keywords: trace codes
zbMATH Keywords: fibre product of Artin-Schreier curves
zbMATH Keywords: rational points
zbMATH Keywords: supersingular curves
reviewed by: Amílcar Pacheco
MaRDI profile type: Publication
full work available at URL: 0097-3165(95)90099-3
OpenAlex ID: W1989882541
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description: scientific article; zbMATH DE number 753624