In this paper, for an odd prime p such that p≡3 mod 4, odd n, and d=(pⁿ+1)/(p^k+1)+(pⁿ-1)/2 with k|n, the value distribution of the exponential sum S(a,b) is calculated as a and b run through $\mathbb{F}_{p^n}$. The sequence family $\mathcal{G}$ in which each sequence has the period of N=pⁿ-1 is also constructed. The family size of $\mathcal{G}$ is pⁿ and the correlation magnitude is roughly upper bounded by $(p^k+1)\sqrt{N}/2$. The weight distribution of the relevant cyclic code C over $\mathbb{F}_p$ with the length N and the dimension ${\rm dim}_{\mathbb{F}_p}\mathcal{C}=2n$ is also derived.