Let \(q=2^m, m\in\mathbb{N},\) an \([n,k,d]_q\) binary linear code \(\mathcal C\) is a \(k\)-dimensional subspace of \(\mathbb{F}_q^n\) with minimal Hamming distance \(d.\) The code cyclic if \((c_0,\ldots,c_{n-1})\in\mathcal C\) implies \((c_{n-1}, c_0,\ldots,c_{n-2})\in\mathcal C.\)
This work explores a new class of binary cyclic codes \(\mathcal C_{(u,v)}\) with parameters \([2^m-1, 2^m-m-9, 4],\) where \(m = 8k\) for a positive integer \(k\ge 2\) and \((u, v) = (1, \frac{2^m-1}{17})\) with two nonzeros \(\alpha^{-u}\) and \(\alpha^{-v}.\) The value distribution of \(S_{u,v}(a, b)\) is calculated. This is done by considering four cases: (1) \((a, b) = (0, 0);\) (2) \((a, 0)\in \mathbb{F}_{2^m}^\ast\times\{0\};\) (3) \((0, b) \in\{0\}\times \mathbb{F}_{2^8}^\ast;\) (4) \((a, b)\in \mathbb{F}_{2^m}^\ast\times\mathbb{F}_{2^8}^\ast.\) For this particular case, it is also shown that the code \(\mathcal C_{(u,v)}\) is distance-optimal with respect to the sphere packing bound.
The weight distribution of the dual of \(\mathcal C_{(u,v)}\) is completely determined based on some results on Gaussian periods.