On the weight distributions of several classes of cyclic codes from APN monomials

Abstract: Let $m\geq 3$ be an odd integer and $p$ be an odd prime. % with $p-1=2<sup>rh$, where $h$ is an odd integer. In this paper, many classes of three-weight cyclic codes over $\mathbb{F}<em>{p}$ are presented via an examination of the condition for the cyclic codes $\mathcal{C}</em>{(1,d)}$ and $\mathcal{C}<em>{(1,e)}$, 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 $\pi<sup>{-i}$ over $\mathbb{F}</em>{p}$ for a primitive element $\pi$ of $\mathbb{F}<em>{p<sup>m}$. %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 $$(p^k+1)\cdot e\equiv 2 p<sup>{\tau}\pmod{p<sup>m-1}$$, the value distributions of the two exponential sums $T(a,b)=\sum\limits</em>{x\in \mathbb{F}<em>{p<sup>m}}\omega<sup>{\Tr(ax+bx<sup>e)}$ and $ S(a,b,c)=\sum\limits</em>{x\in \mathbb{F}<em>{p^{m}}\omega^{{\Tr(ax+bx^{e+cx^s)},}}} $ where $s=(p<sup>m-1)/2$, are settled. As an application, the value distribution of $S(a,b,c)$ is utilized to investigate the weight distribution of the cyclic codes $\mathcal{C}</em>{(1,e,s)}$ 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.