Self-Dual Abelian Codes over Galois Rings
Somphong Jitman, San Ling · arXiv (Cornell University) · 2014
In this paper, we study Euclidean and Hermitian self-dual abelian codes in ${\rm GR}(p^r,s)[G]$, a group ring of a finite abelian group $G$ over a Galois ring ${\rm GR}(p^r,s)$. Characterizations of such self-dual codes are given together with necessary and sufficient conditions for the existence of a Euclidean self-dual abelian code or a Hermitian self-dual abelian code in ${\rm GR}(p^r,s)[G]$. The enumeration formulas of such self-dual codes are established. However, the formulas contain terms which have not been well studied. In the case where $\gcd(|G|,p)=1$, we determine explicitly the numbers of Euclidean and Hermitian self-dual abelian codes in ${\rm GR}(p^r,s)[G]$. Applying known results on cyclic codes of length $p^a$ over ${\rm GR}(p^2,s)$, we arrive at explicit formulas for the numbers of Euclidean and Hermitian self-dual abelian codes in ${\rm GR}(p^2,s)[G]$, where the Sylow $p$-subgroup of $G$ is cyclic. The analog results for Euclidean and Hermitian self-dual cyclic codes over Galois rings are therefore obtained as corollaries.