AMELIORATION OF THE McWILLIAMS.SLOANE TABLES USING GEOMETRIC CODES FROM CURVES WITH GENUS 1,2 OR 3. (following A.M.Barg,G.L.Katsman and M.A.Tsfasman)
Yves Driencourt, Jean Francis Michon · 2005
The theory of geometric codes inaugurated by Goppa ([3],[4]) has already led to several decisive results in coding theory, whose the best known is the construction of an infinite family of codes with parameters better than the Varshamov-Gilbert bound (work of Tsfasman ,Vladut et Zink ,[11],[12]). Beside the study of new families of codes (issued from elliptic curves for example) there exist some less known results showing the power of the geometric tools : among others the amelioration of the table concerning the best known binary codes (of length up to 512 and minimum distance up to 29) appearing in the book by F.J. McWilliams and N.J.A. Sloane ([8]) and the construction of sphere packings associated with geometric codes .The aim of this paper is to illustrate the first of these topics. To do this, we build a few codes whose parameters are announced in the work of Barg , Katsman and Tsfasman ([1 ]). These codes are obtained in the following way : one considers outer geometric codes over F 8 ou F16 concatenated with inner binary codes as shown by Zinoviev