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

Abstract

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,e) over Fp, which have parity-check polynomial m1(x)me (x), are presented by examining general conditions on the parameters p, m and e, where mi (x) is the minimal polynomial of π−i over Fp for a primitive element π of Fpm . Furthermore, for p ≡ 3 (mod 4) and a positive integer e satisfying (pk + 1) · e ≡ 2 (mod pm − 1) for some positive integer k with gcd(m, k) = 1, the value distributions of the exponential sums T(a, b) = ∑ x∈Fpm ωTr(ax+bxe ) and S(a, b, c) = ∑ x∈Fpm ωTr(ax+bxe +cxs ), where s = (pm − 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 m1(x)me (x)ms (x). In the case of p = 3 and even e satis- fying the above condition, the dual of the cyclic code C(1,e,s) has optimal minimum distance.