Characterization and enumeration of complementary dual abelian codes

Abstract

Abelian codes and complementary dual codes form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, a family of abelian codes with complementary dual in a group algebra F p ν [G] has been studied under both the Euclidean and Hermitian inner products, where p is a prime, ν is a positive integer, and G is an arbitrary finite abelian group. Based on the discrete Fourier transform decomposition for semi-simple group algebras and properties of ideas in local group algebras, the characterization of such codes have been given. Subsequently, the number of complementary dual abelian codes in F p ν [G] has been shown to be independent of the Sylow p-subgroup of G and it has been completely determined for every finite abelian group G. In some cases, a simplified formula for the enumeration has been provided as well. The known results for cyclic complementary dual codes can be viewed as corollaries.

Self-conjugate-reciprocal irreducible monic factors of x − 1 over finite fields and their applications

“…First, we note that the characterization and enumeration of Hermitian complementary dual cyclic codes over finite fields can be viewed as the special case of Hermitian complementary dual abelian codes in group algebras [3], where the underlying group is cyclic. Here, simpler and direct study of such complementary dual codes of arbitrary lengths using SCRIM factors of x n − 1 in F q 2 [x].…”

Self-dual and complementary dual abelian codes over Galois rings

“…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], [3], [11], [13], [15], [21], [26], [27], 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].…”

“…Euclidean complementary dual cyclic codes over finite fields have been studied in [27]. Recently, they have been generalized to Euclidean and Hermitian complementary dual abelian codes over finite fields in [3]. The complete characterization and enumeration of complementary dual abelian codes over finite fields have been established in the said paper.…”

On constacyclic codes and their generalizations

“…As mentioned in Subsection 5.1.2, for m = 0, generalized negacyclic codes become abelian codes in group algebras. The study of self-dual and complementary dual abelian codes is given in [11], [30] and [31]. Throughout this section, assume that m ≥ 1.…”