Xiwang Cao — Page 3

arXiv2010

On Exponential Sums, Nowton identities and Dickson Polynomials over Finite Fields

Let $\mathbb{F}_{q}$ be a finite field, $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with $\gcd(n,q)=1$. We present a recursive formula for evaluating the exponential sum $\sum_{c\in \mathbb{F}_{q^s}}χ^{(s)}(f(x))$. Let $a$ and $b$ be two elements in $\mathbb{F}_q$ with $a\neq 0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $\sum_{c\in \mathbb{F}^*_{q^s}}χ^{(s)}(ac^u+bc^{-1})$, where $χ^{(s)}$ is the lifting of an additive character $χ$ of $\mathbb{F}_q$. Some properties of the sequences constructed from these exponential sums are provided also.

cs.IT↗

arXiv2009

Weight Distribution of A p-ary Cyclic Code

For an odd prime $p$ and two positive integers $n\geq 3$ and $k$ with $\frac{n}{{\rm gcd}(n,k)}$ being odd, the paper determines the weight distribution of a $p$-ary cyclic code $\mathcal{C}$ over $\mathbb{F}_{p}$ with nonzeros $α^{-1}$, $α^{-(p^k+1)}$ and $α^{-(p^{3k}+1)}$, where $α$ is a primitive element of $\mathbb{F}_{p^n}$

cs.IT↗