Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Left dihedral codes over Galois rings ${\rm GR}(p^2,m)$   Let $D_{2n}=\langle x,y\mid x^n=1, y^2=1, yxy=x^{-1}\rangle$ be a dihedral
group, and $R={\rm GR}(p^2,m)$ be a Galois ring of characteristic $p^2$ and
cardinality $p^{2m}$ where $p$ is a prime. Left ideals of the group ring
$R[D_{2n}]$ are called left dihedral codes over $R$ of length $2n$, and
abbreviated as left $D_{2n}$-codes over $R$. Let ${\rm gcd}(n,p)=1$ in this
paper. Then any left $D_{2n}$-code over $R$ is uniquely decomposed into a
direct sum of concatenated codes with inner codes ${\cal A}_i$ and outer codes
$C_i$, where ${\cal A}_i$ is a cyclic code over $R$ of length $n$ and $C_i$ is
a skew cyclic code of length $2$ over an extension Galois ring or principal
ideal ring of $R$, and a generator matrix and basic parameters for each outer
code $C_i$ is given. Moreover, a formula to count the number of these codes is
obtained, the dual code for each left $D_{2n}$-code is determined and all
self-dual left $D_{2n}$-codes and self-orthogonal left $D_{2n}$-codes over $R$
are presented, respectively.

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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: eess
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The categories of this paper are: hep-ex