Estimation asymptotique de la distance minimale du dual des codes BCH et polynômes de Dickson
✍ Scribed by François Rodier
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 506 KB
- Volume
- 149
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
✦ Synopsis
We discuss the Carlitz-Uchiyama bound for the duals of BCH codes. An improvement of this bound was expected by MacWilliams and Sloane in their book The Theory of Error-Correctiny Codes. We show that there exists a conjecturally infinite series of values such that for the designed distance of a BCH code belonging to that series this improvement is not true. On the contrary, we show that the Carlitz-Uchiyama bound is reached asymptotically. We deduce these results from corresponding results for exponential sums associated to monomials or to Dickson polynomials. We study also the extension of these results to the case of fields of characteristic different from 2.
R~um~
On discute de la borne de Carlitz Uchiyama pour le dual d'un code BCH. MacWilliams et Sloane avaient sugg6r6 dans leur livre The Theory of Error-Correctin9 Codes une am61ioration de cette borne. On montre, pour la distance prescrite du code BCH appartenant fi une certaine s6rie, s6rie qui est conjecturalement infinie, que cette amelioration n'est pas verifi6e et qu'au contraire la borne de Carlitz-Uchiyama est asymptotiquement atteinte. On d6duit ces r6sultats de r6sultats correspondants pour les sommes exponentielles associ6es fi des mon6mes ou fi des polyn6mes de Dickson. On 6tudie aussi l'extension de ces r6sultats au cas des corps de caract6ristique sup~rieure /l 2.