Handbook of theoretical computer science (vol. A)
New bounds on cyclic codes from algebraic curves
Proceedings of the 3rd International Colloquium on Coding Theory and Applications
Computer algebra handbook
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Let p be an odd prime and \zeta_p be a primitive pth root of unity over \smallBbb{Q}. The Galois group G of K:=\smallBbb{Q}(\zeta_p) over \smallBbb{Q} is a cyclic group of order p-1. The integral group ring \smallBbb{Z}[G] contains the Stickelberger ideal S_p which annihilates the ideal class group of K. In this paper we investigate the parameters of cyclic codes S_p(q) obtained as reductions of S_p modulo primes q which we call Stickelberger codes. In particular, we show that the dimension of S_p(p) is related to the index of irregularity of p, i.e., the number of Bernoulli numbers B_{2k}, 1\le k\le (p-3)/2, which are divisible by p. We then develop methods to compute the generator polynomial of S_p(p). This gives rise to anew algorithm for the computation of the index of irregularity of a prime. As an application we show that 20,001,301 is regular. This significantly improves a previous record of 8,388,019 on the largest explicitly known regular prime.