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, many classes of three-weight cyclic codes over Fp are presented via an examination of the condition for the cyclic codes C(1,d) and C(1,e), which have parity-check polynomials m1(x)md(x) and m1(x)me(x) respectively, to have the same weight distribution, 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 positive integers e such that there exist integers k with gcd(m, k) = 1 and τ ∈ {0, 1,··· , m− 1} satisfying (p k + 1)· e ≡ 2p τ (mod p m − 1), the value distributions of the two exponential sums T(a, b) = ∑ x∈Fpm ω Tr(ax+bx e ) and S(a, b, c) = ∑ x∈Fpm ω Tr(ax+bx e +cx s ) , where s = (p m − 1)/2, are settled. As an application, the value distribution of S(a, b, c) is utilized to investigate 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 satisfying the above condition, the duals of the cyclic code s C(1,e,s) have the optimal minimum distance.