Exponential Sums, Cyclic Codes and Sequences: the Odd Characteristic Kasami Case

Abstract

Let $q=p^n$ with $n=2m$ and $p$ be an odd prime. Let $0\le k\le n-1$ and $k\ne m$. In this paper we determine the value distribution of following exponential(character) sums \[\sum\limits_{x\in \bF_q}ζ_p^{\Tra_1^m (αx^{p^{m}+1})+\Tra_1^n(βx^{p^k+1})}\quad(α\in \bF_{p^m},β\in \bF_{q})\] and \[\sum\limits_{x\in \bF_q}ζ_p^{\Tra_1^m (αx^{p^{m}+1})+\Tra_1^n(βx^{p^k+1}+\ga x)}\quad(α\in \bF_{p^m},β,\ga\in \bF_{q})\] where $\Tra_1^n: \bF_q\ra \bF_p$ and $\Tra_1^m: \bF_{p^m}\ra\bF_p$ are the canonical trace mappings and ${\zeta }_p=e^{\frac{2\pi i}{p}}$ is a primitive $p$-th root of unity. As applications: (1). We determine the weight distribution of the cyclic codes $\cC_1$ and $\cC_2$ over $\bF_{p^t}$ with parity-check polynomials $h_2(x)h_3(x)$ and $h_1(x)h_2(x)h_3(x)$ respectively where $t$ is a divisor of $d=\gcd (m,k)$, and $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of ${\pi }^{-1}$, ${\pi }^{-(p^k+1)}$ and ${\pi }^{-(p^m+1)}$ over $\bF_{p^t}$ respectively for a primitive element $\pi$ of $\bF_q$. (2). We determine the correlation distribution among a family of m-sequences. This paper extends the results in \cite{Zen Li}.

References in corpus (1)

• A Class of Nonbinary Codes and Their Weight Distribution

Cited by in corpus (2)

• Cyclic Codes and Sequences from a Class of Dembowski-Ostrom Functions
• Cyclic Codes and Sequences: the Generalized Kasami Case