Abstract
Let p be an odd prime and \(\zeta _p\) be a primitive p th root of unity over \({\mathbb{Q}}\). The Galois group G of \(K: = {\mathbb{Q}}(\zeta _p )\) over \({\mathbb{Q}}\) is a cyclic group of order p-1. The integral group ring \({\mathbb{Z}}\)[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 , \(1 \leqslant k \leqslant (p - 3)/2\), 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.
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Shokrollahi, M.A. Stickelberger Codes. Designs, Codes and Cryptography 9, 203–213 (1996). https://doi.org/10.1023/A:1018022215278
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DOI: https://doi.org/10.1023/A:1018022215278