Self-dual and complementary dual abelian codes over Galois rings

Year 2019, Volume: 6 Issue: 2, 75 - 94, 07.05.2019

Somphong Jitman San Ling

https://doi.org/10.13069/jacodesmath.560406

Abstract

Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications.
In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring ${ GR}(p^r,s)[G]$, where $G$ is a finite abelian group and ${ GR}(p^r,s)$ is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in ${ GR}(p^r,s)[G]$. A general formula for the number of such self-dual codes is established. In the case where $\gcd(|G|,p)=1$, the number of self-dual abelian codes in ${ GR}(p^r,s)[G]$ is completely and explicitly determined.
Applying known results on cyclic codes of length $p^a$ over ${ GR}(p^2,s)$, an explicit formula for the number of self-dual abelian codes in ${ GR}(p^2,s)[G]$ are given, where the Sylow $p$-subgroup of $G$ is cyclic.
Subsequently, the characterization and enumeration of complementary dual abelian codes in ${ GR}(p^r,s)[G]$ are established.
The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.

Keywords

Abelian codes, Galois rings, Self-dual codes, Complementary dual codes, Codes over rings

Thanks

S. Jitman was supported by the Thailand Research Fund and Silpakorn University under Research Grant RSA6280042. S. Ling was supported by Nanyang Technological University Research Grant M4080456.

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