Kanonische Codes auf einigen Überdeckungskurven
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.
scientific article; zbMATH DE number 168960