Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the weight distributions of several classes of cyclic codes from APN
  monomials   Let $m\geq 3$ be an odd integer and $p$ be an odd prime. % with $p-1=2^rh$,
where $h$ is an odd integer.
  In this paper, many classes of three-weight cyclic codes over
$\mathbb{F}_{p}$ are presented via an examination of the condition for the
cyclic codes $\mathcal{C}_{(1,d)}$ and $\mathcal{C}_{(1,e)}$, which have
parity-check polynomials $m_1(x)m_d(x)$ and $m_1(x)m_e(x)$ respectively, to
have the same weight distribution, where $m_i(x)$ is the minimal polynomial of
$\pi^{-i}$ over $\mathbb{F}_{p}$ for a primitive element $\pi$ of
$\mathbb{F}_{p^m}$. %For $p=3$, the duals of five classes of the proposed
cyclic codes are optimal in the sense that they meet certain bounds on linear
codes. Furthermore, for $p\equiv 3 \pmod{4}$ and positive integers $e$ such
that there exist integers $k$ with $\gcd(m,k)=1$ and $\tau\in\{0,1,\cdots,
m-1\}$ satisfying $(p^k+1)\cdot e\equiv 2 p^{\tau}\pmod{p^m-1}$, the value
distributions of the two exponential sums $T(a,b)=\sum\limits_{x\in
\mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e)}$ and $ S(a,b,c)=\sum\limits_{x\in
\mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e+cx^s)}, $ where $s=(p^m-1)/2$, are
settled. As an application, the value distribution of $S(a,b,c)$ is utilized to
investigate the weight distribution of the cyclic codes $\mathcal{C}_{(1,e,s)}$
with parity-check polynomial $m_1(x)m_e(x)m_s(x)$. In the case of $p=3$ and
even $e$ satisfying the above condition, the duals of the cyclic codes
$\mathcal{C}_{(1,e,s)}$ have the optimal minimum distance.

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The categories of this paper are: nucl-ex
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas