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.
Highlights
Structured codes over finite fields with self-duality and complementary duality are important families of linear codes that have been extensively studied for both theoretical and practical reasons
Abelian codes over Galois rings are studied in terms of the ideals in the group ring GR(pr, s)[G], where G is a finite abelian group and GR(pr, s) is a Galois ring
We focus on abelian codes over Galois rings GR(pr, s), i.e., ideals in the group ring GR(pr, s)[G] of an abelian group G over a Galios ring GR(pr, s)
Summary
Structured codes over finite fields with self-duality and complementary duality are important families of linear codes that have been extensively studied for both theoretical and practical reasons (see [1], [3], [11], [13], [15], [21], [26], [27], and references therein). In [1], [5], [4] and [23], characterization and enumeration of Euclidean and Hermitian self-dual cyclic codes over finite chain rings have been discussed. The complete characterization and enumeration of complementary dual abelian codes over finite fields have been established in the said paper. We study self-dual and complementary dual abelian codes in GR(pr, s)[G] with respect to both the Euclidean and Hermitian inner products. The complete enumeration of Euclidean and Hermitian self-dual abelian codes in GR(pr, s)[G] is given in the special cases where i) gcd(p, |G|) = 1; and ii) r = 2 and the Sylow p-subgroup of G is cyclic.
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