Self-dual and complementary dual abelian codes over Galois rings

• Main Authors: Jitman, Somphong, Ling, San
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2020
• Subjects: Science::Mathematics; Abelian Codes; Galois Rings
• Online Access: https://hdl.handle.net/10356/142098
• Institution: Nanyang Technological University

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(pr,s)[G], where G is a finite abelian group and GR(pr,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(pr,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(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 for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.