Poids des duaux des codes BCH de distance prescrite 2 a +1 et sommes exponentielles  [ Weight of duals of BCH codes of designed distance 2 a +1 and exponential sums ]
Bulletin de la Société Mathématique de France, Volume 130 (2002) no. 2, p. 211-231

Let n be an even integer. We consider a binary BCH code C n of length 2 n -1 and designed distance 2 a +1 with a3. The weight of a nonzero codeword of the dual of C n is linked to the value of an exponential sum. We will show that this exponential sum does not reach the Weil bound and we will improve this bound. Thus, we obtain an improvement of the Carlitz-Uchiyama bound on the weights of the words of the dual of C n .

Soit n un entier pair. On considère un code BCH binaire C n de longueur 2 n -1 et de distance prescrite 2 a +1 avec a3. Le poids d’un mot non nul du dual de C n peut s’exprimer en fonction d’une somme exponentielle. Nous montrerons que cette somme n’atteint pas la borne de Weil et nous proposerons une amélioration de celle-ci. En conséquence, nous obtiendrons une amélioration de la borne de Carlitz-Uchiyama sur le poids des mots du dual de C n .

DOI : https://doi.org/10.24033/bsmf.2418
Classification:  11T23,  94B15
Keywords: BCH codes, Carlitz-Uchiyama bound, exponential sums, Weil bound
@article{BSMF_2002__130_2_211_0,
     author = {F\'erard, \'Eric},
     title = {Poids des duaux des codes BCH de distance prescrite $2^a+1$ et sommes exponentielles},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {130},
     number = {2},
     year = {2002},
     pages = {211-231},
     doi = {10.24033/bsmf.2418},
     zbl = {1028.11074},
     mrnumber = {1924541},
     language = {fr},
     url = {http://www.numdam.org/item/BSMF_2002__130_2_211_0}
}
Poids des duaux des codes BCH de distance prescrite $2^a+1$ et sommes exponentielles. Bulletin de la Société Mathématique de France, Volume 130 (2002) no. 2, pp. 211-231. doi : 10.24033/bsmf.2418. http://www.numdam.org/item/BSMF_2002__130_2_211_0/

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