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,e) over F_p, which have parity-check polynomial m₁(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 π-i over Fp for a primitive element π of F_pm. Furthermore, for p ≡ 3 (mod 4) and a positive integer e satisfying (p^k + 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) = Σx∈_Fpm ω^Tr(ax+bxe) and S(a, b, c) = Σx∈_Fpm ωTr(ax+bx^e+cx^s), 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 m₁(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,e,s) has optimal minimum distance.