Quadratic double circulant codes over fields
The author generalizes and uniformizes some constructions of double circulant codes using quadratic residues. The general construction leads to what he calls QDC (quadratic double circulant) codes. Depending on the characteristic of the field over which the codes are defined, the construction contains and generalizes results by \textit{M. Karlin} [IEEE Trans. Inf. Theory 15, 81-92 (1969; Zbl 0167.18104)] for characteristic 2, \textit{G. F. M. Beenker} [ibid. 30, 403-405 (1984; Zbl 0541.94018)] and \textit{E. Dawson} [Ars Comb. 19 A, 303-308 (1985; Zbl 0558.94011)] for characteristic 3, \textit{J. Leon}, \textit{V. Pless} and \textit{N. Sloane} [J. Comb. Theory, Ser. A 32, 178-194 (1982; Zbl 0483.94018)] for characteristic 5, and improves results of \textit{V. Pless} and \textit{V. Tonchev} [IEEE Trans. Inf. Theory 33, 723-727 (1987; Zbl 0658.94016)] in the sense that the minimum distance of the newly constructed codes is higher given the other parameters. It is a pity that the author gives no general results on the minimum weight for the constructed codes. Instead of this he computed the minimum weight for low characteristics and moderate choices of other parameters, using the magma package. Nevertheless he improves on several of the results from the tables of \textit{A. Brouwer} [Handbook of coding theory (Elsevier; Amsterdam) (1998; Zbl 0936.94015)]. Some of the new results are even optimal according to these tables. Furthermore some results are given on automorphism groups of the constructed codes. The paper is a good basis for further research on QDC codes.
scientific article; zbMATH DE number 1763795
scientific article