Self-dual and complementary dual abelian codes over Galois rings

Journal of Algebra Combinatorics Discrete Structures and Applications

article

Somphong Jitman¹  San Ling² 

[1] Department of Mathematics, Faculty of Science, Silpakorn University;Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University

关键词: Abelian codes;    Galois rings;    Self-dual codes;    Complementary dual codes;    Codes over rings;   

DOI  :  10.13069/jacodesmath.560406

学科分类：社会科学、人文和艺术（综合）

来源: Yildiz Technical University

【 摘 要 】

Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes oflinear 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 groupring GR(pr, s)[G], where G is a finite abelian group and GR(pr, s) is a Galois ring. Characterizationsof self-dual abelian codes have been given together with necessary and sufficient conditions for theexistence of a self-dual abelian code in GR(pr, s)[G]. A general formula for the number of such selfdual codes is established. In the case where gcd(|G|, p) = 1, the number of self-dual abelian codesin GR(pr, s)[G] is completely and explicitly determined. Applying known results on cyclic codes oflength paover GR(p2, s), an explicit formula for the number of self-dual abelian codes in GR(p2, 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(pr, s)[G] are established. The analogous results forself-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.

【 授权许可】

CC BY