The Weight Distributions of Several Classes of Cyclic Codes From APN Monomials

Let m >= 3 be an odd integer and p be an odd prime. In this paper, a number of classes of three-weight cyclic codes C-(1,C-e) over F-p, which have parity-check polynomial m(1)(x)m(e)(x), are presented by examining general conditions on the parameters p, m, and e, where m(i)(x) is the minimal polynomial of pi(-i) over F-p for a primitive element pi of F(p)m. Furthermore, for p equivalent to 3 (mod 4) and a positive integer e satisfying (p(k) + 1) . e equivalent to 2 (mod p(m) - 1) for some positive integer k with gcd(m, k) = 1, the value distributions of the exponential sums T(a, b) = Sigma(x is an element of)F(p)m omega(Tr(ax+bxe)) and S(a, b, c) = Sigma(x is an element of)F(p)m omega(Tr(ax+bxe+cxs)), where s = (p(m) - 1)/2, are determined. As an application, the value distribution of S(a, b, c) is utilized to derive the weight distribution of the cyclic codes C((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 dual of the cyclic code C-(1,C- e,C- s) has optimal minimum distance.