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

In this paper, for an odd prime $p$ such that $p\equiv 3\bmod 4$, odd $n$, and $d=(p^{n+1)/(pk+1)+(pn-1)/2$} 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 $N=p^n-1$ is also constructed. The family size of $\mathcal{G}$ is $p^n$ and the correlation magnitude is roughly upper bounded by $(p^{k+1)\sqrt{N}/2$.} The weight distribution of the relevant cyclic code $\mathcal{C}$ over $\mathbb{F}p$ with the length $N$ and the dimension ${\rm dim}{\mathbb{F}p}\mathcal{C}=2n$ is also derived. Our result includes the case in \cite{Xia} as a special case.