# American Institute of Mathematical Sciences

## Journals

AMC

Let $p$ be an odd prime, $n≥q3$ and $k$ positive integers with $e=\gcd(n,k)$. In this paper, a new family $\mathcal{S}$ of $p$-ary sequences with period $N=p^n-1$ is proposed. The sequences in $\mathcal{S}$ are constructed by adding a $p$-ary sequence to its two decimated sequences with different phase shifts. The correlation distribution among sequences in $\mathcal{S}$ is completely determined. It is shown that the maximum magnitude of nontrivial correlations of $\mathcal{S}$ is upper bounded by $p^e\sqrt{N+1}+1$, and the family size of $\mathcal{S}$ is $N^2$. Our sequence family has a large family size and low correlation.

keywords: Sequence family m-sequence cross-correlation exponential sum
AMC

For two odd integers $l,k$ with $0<l<k$ and $\gcd(l,k)=1$, let $m=2k$ and $d=\frac{2^{lk}+1}{2^l+1}$. In this paper, we study the cross-correlation between a binary $m$-sequence $(s_t)$ of length $2^m-1$ and its $d$-decimated sequences $(s_{dt+u}), 0≤q u<\frac{2^k+1}{3}.$ It is shown that the maximum magnitude of cross-correlation values is $2^{\frac{m}{2}+1}+1.$ Moreover, a new sequence family with maximum correlation magnitude $2^{\frac{m}{2}+1}+1$ and family size $2^{\frac{m}{2}}$ is proposed.

keywords: Binary sequence cross-correlation exponential sum sequence family
AMC
Let $p$ be an odd prime, $n=2m$, and $n/\gcd(k,n)$ be odd. In this paper, we study the cross correlation between a $p$-ary $m$-sequence $(s_{t})$ of period $p^{n}-1$ and its decimated sequence $(s_{dt})$ where $d$ satisfies $d(p^k+1)\equiv p^m+1 \pmod {p^n-1}$. Our results show that the cross-correlation function is six-valued and the distribution of the cross correlation is also completely determined.
keywords: $p$-ary sequence exponential sum Cross correlation quadratic form.