Further improvements on the designed minimum distance of algebraic geometry codes
type: Item
instance of: scholarly article
title: Further improvements on the designed minimum distance of algebraic geometry codes
zbMATH Open document ID: 1160.14012
author: Cem Güneri
author: Henning Stichtenoth
author: Ịhsan Taşkın
published in: Journal of Pure and Applied Algebra
publication date: 2008-11-11T00:00:00Z
review text: Let \(F\) be an algebraic function field of genus \(g\) with full constant field \(\mathbb{F}_q\), where \(q\) is a prime power. Let \(G\) and \(D\) be two divisors of \(F\) with disjoint support and such that \(D\) is the sum of \(n\) different rational places. The residue Goppa code \(C_{\Omega}=C_{\Omega}(D,G)\) has length \(n\), dimension \(i(G-D)-i(G)\), and minimal distance \(d(C_{\Omega})\geq \deg G-(2g-2)\). There are a lot of results in the literature, improving this designed minimum distance. In the paper under review, the authors synthesize most of these results in two theorems that take the form: \[ d(C_{\Omega})- \deg G+(2g-2)\geq \deg Z+\left(i(A)-i(G-C)\right),\;\text{ or }\geq\deg Z+1, \] where \(A,C,Z\) are divisors satisfying opportune conditions. Most of the known improvements of the Goppa bound for the minimum distance are particular cases of these theorems. Also, some examples are discussed where these bounds improve on the previous ones. The ideas behind these improvements of the designed minimum distance are often based on \textit{shrinking} or \textit{growing} the divisors while keeping fixed their attached Riemann-Roch vector spaces. In this regard, the authors include in the paper a general analysis of the equivalence classes of divisors determined by the equivalence relation: \(M\approx N \Longleftrightarrow\mathcal{L}(M)=\mathcal{L}(N)\).
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 5364858
zbMATH Keywords: Goppa codes
zbMATH Keywords: designed minimum distance, floor of a divisor
reviewed by: Enric Nart
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.05.003
OpenAlex ID: W1992003595
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DOI: 10.1016/J.JPAA.2008.05.003
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description: scientific article; zbMATH DE number 5364858