Kanonische Codes auf einigen Überdeckungskurven
type: Item
instance of: scholarly article
title: Kanonische Codes auf einigen Überdeckungskurven
zbMATH Open document ID: 0778.94009
DOI: 10.1007/BF02567060
author: B. Heinrich Matzat
published in: Manuscripta Mathematica
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/155785
review text: First the author gives simpler proofs for some of the results of \textit{R. Pellikaan}, \textit{B. Z. Shen} and \textit{G. J. M. van Wee}, IEEE Trans. Inform. Theory 37, No. 3, 583-602 (1991). Then he gives some results on the dimension of certain canonic codes over the function field on top of an Artin-Schreier function field chain. Furthermore it is proven that the dual of these codes is also canonic, more specific the codes appear to be self-orthogonal and for the right dimension and in characteristic 2 self dual. The notion of genus of a code is introduced and genus and distance are calculated for canonic codes from Artin-Schreier function fields. Also some asymptotic implications are looked upon. Moreover as an example the author constructs \([q^ m,q^ m-(m+1),3]\) codes over \(GF(q)\) and by concatenating these codes in the case \(q=4\) with an [2,2,1] code he obtains linear codes over \(GF(2)\), having parameters \([2^{2m+1},2^{2m+1}-2(m+1),3]\). He remarks that in the tables of Sloane and MacWilliams there are no better linear codes with the same length and distance. According to me this is quite explainable since this dimension is best possible according to the sphere packing bound. Finally the author compares the parameters of the canonic Artin-Schreier codes with those of the codes obtained from Hermitean Curves. In some sense the parameters of Artin-Schreier codes are better.
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 168960
zbMATH Keywords: algebraic-geometric codes
zbMATH Keywords: Artin-Schreier curves
zbMATH Keywords: Hermitean curves
zbMATH Keywords: linear codes over \(GF(2)\)
zbMATH Keywords: Artin-Schreier function field
zbMATH Keywords: canonic codes
zbMATH Keywords: sphere packing bound
reviewed by: Herman J. Tiersma
MaRDI profile type: Publication
cites work: Q3994524
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description: scientific article; zbMATH DE number 168960