Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings

Abstract

Let R be the Galois ring GR(pk,s) of characteristic pk and cardinality psk. Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length n over R, and a p-adic integer ring with gcd(p,n)=1. Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length rlm over R, where r,l, and t are three primes with 2∤l, r|(qt−1), lv∥(qt−1) and gcd(rl,q(q−1))=1. Finally, as applications, weight distributions of all irreducible cyclic codes for t=2 and generator polynomials of self-dual cyclic codes of length lm and rlm over R are given.