Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Characterization and Enumeration of Complementary Dual Abelian Codes   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 $\mathbb{F}_{p^\nu}[G]$ has been studied
under both the Euclidean and Hermitian inner products, where $p$ is a prime,
$\nu$ 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 $\mathbb{F}_{p^\nu}[G]$ 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.

25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
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The categories of this paper are: supr-con