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 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 π-i over Fp for a primitive element π of F
_p
m. 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∈
_F
p
m
ω
^Tr(ax+bx
e
)
and S(a, b, c) = Σx∈
_F
p
m
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 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,e,s)
has optimal minimum distance.