Self-dual and complementary dual abelian codes over Galois rings

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. rings 2010 AMS Classification. 94B15, 94B05, 16A261 Introduction Algebraically 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], [5], [11], [13], [15], [20], [25], [26], and references therein). Codes over finite rings have been interesting since it was proven that some binary non-linear codes such as the Kerdock, Preparata, and Goethal codes are the Gray images of linear codes over Z 4 [10]. Algebraically structured codes such as cyclic, constacyclic, and abelian codes have extensively been studied over Z p r , Galois rings, and finite chain rings in general (see [7],[18], and references therein). The characterization and enumeration of Euclidean self-dual cyclic codes over finite fields have been established in [11] and generalized to Euclidean and Hermitian self-dual abelian codes over finite fields in [13] and [15], respectively. Over some finite rings, a characterization of self-dual cyclic, constacyclic and abelian codes has been done (see, for example, [1], [7],[16], [17], [23], and S. Jitman (Corresponding Author) is with the ). S. Ling is with the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, REPUBLIC of SINGAPORE (lingsan@ntu.edu.sg).[25]). In [1], [3], [5] and [22], characterization and enumeration of Euclidean and Hermitian self-dual cyclic codes over finite chain rings have been discussed. Euclidean complementary dual cyclic codes over finite fields have been studied in [26]. Recently, they have been generalized to Euclidean and Hermitian complementary dual abelian codes over finite fields in [5]. The complete characterization and enumeration of compleme...