The Walsh transform of a class of monomial functions and cyclic codes

Abstract

Let 𝔽 _p be a finite field with p elements, where p is a prime. Let N ≥ 2 be an integer and f the least positive integer satisfying p ^f ≡ −1 (mod N). Then we let q = p ^2f and r = q ^m. In this paper, we study the Walsh transform of the monomial function

$f(x)=\text {Tr}_{r/p}(ax^{\frac {r-1} N})$

for $a \in \Bbb F_{r}^{*}$. We shall present the value distribution of the Walsh transform of f(x) and show that it takes at most $\min \{p, N\}+1$ distinct values. In particular, we can obtain binary functions with three-valued Walsh transform and ternary functions with three-valued or four-valued Walsh transform. Furthermore, we present two classes of four-weight binary cyclic codes and six-weight ternary cyclic codes.