Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On {\sigma}-LCD codes   Linear complementary pairs (LCP) of codes play an important role in armoring
implementations against side-channel attacks and fault injection attacks. One
of the most common ways to construct LCP of codes is to use Euclidean linear
complementary dual (LCD) codes. In this paper, we first introduce the concept
of linear codes with $\sigma$ complementary dual ($\sigma$-LCD), which includes
known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes.
  As Euclidean LCD codes, $\sigma$-LCD codes can also be used to construct LCP
of codes. We show that, for $q > 2$, all q-ary linear codes are $\sigma$-LCD
and that, for every binary linear code $\mathcal C$, the code $\{0\}\times
\mathcal C$ is $\sigma$-LCD. Further, we study deeply $\sigma$-LCD generalized
quasi-cyclic (GQC) codes. In particular, we provide characterizations of
$\sigma$-LCD GQC codes, self-orthogonal GQC codes and self-dual GQC codes,
respectively. Moreover, we provide constructions of asymptotically good
$\sigma$-LCD GQC codes. Finally, we focus on $\sigma$-LCD Abelian codes and
prove that all Abelian codes in a semi-simple group algebra are $\sigma$-LCD.
  The results derived in this paper extend those on the classical LCD codes and
show that $\sigma$-LCD codes allow the construction of LCP of codes more easily
and with more flexibility.

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The categories of this paper are: nlin
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: hep-th