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In this paper, for an odd prime $p$ such that $p\equiv 3\bmod 4$, odd $n$,and $d=(p^n+1)/(p^k+1)+(p^n-1)/2$ with $k|n$, the value distribution of theexponential sum $S(a,b)$ is calculated as $a$ and $b$ run through$\mathbb{F}_{p^n}$. The sequence family $\mathcal{G}$ in which each sequencehas the period of $N=p^n-1$ is also constructed. The family size of$\mathcal{G}$ is $p^n$ and the correlation magnitude is roughly upper boundedby $(p^k+1)\sqrt{N}/2$. The weight distribution of the relevant cyclic code$\mathcal{C}$ over $\mathbb{F}_p$ with the length $N$ and the dimension ${\rmdim}_{\mathbb{F}_p}\mathcal{C}=2n$ is also derived. Our result includes thecase in \cite{Xia} as a special case.

On the Cross-Correlation of a $p$-ary m-Sequence and its Decimated Sequences by $d=\frac{p^n+1}{p^k+1}+\frac{p^n-1}{2}$