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\iteman{io-port 01002527}
\itemau{Shokrollahi, M.A.}
\itemti{Stickelberger codes.}
\itemso{Des. Codes Cryptography 9, No.2, 203-213 (1996).}
\itemab
Summary: Let $p$ be an odd prime and $\zeta_p$ be a primitive $p$th root of unity over $\bbfQ$. The Galois group $G$ of $K:=\bbfQ(\zeta_p)$ over $\bbfQ$ is a cyclic group of order $p-1$. The integral group ring $\bbfZ[G]$ 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 $B_{2k}$, $1\leq k\leq (p-3)/2$, which are divisible by $p$. We then develop methods to compute the generator polynomial of $S_p(p)$. This gives rise to a new 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.
\itemrv{~}
\itemcc{}
\itemut{irregular primes; Stickelberger ideal; cyclic codes}
\itemli{}
\end