4 research outputs found
Self-Dual and Complementary Dual Abelian Codes over Galois Rings
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
, where is a finite abelian group and
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 . A general formula for the
number of such self-dual codes is established. In the case where
, the number of self-dual abelian codes in
is completely and explicitly determined. Applying known results on cyclic codes
of length over , an explicit formula for the number of
self-dual abelian codes in are given, where the Sylow
-subgroup of is cyclic. Subsequently, the characterization and
enumeration of complementary dual abelian codes in are
established. The analogous results for self-dual and complementary dual cyclic
codes over Galois rings are therefore obtained as corollaries.Comment: 22 page