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

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) equivalent to - 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) = Tr-r/p (ax(r-1/N)) for a is an element of 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.