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On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$

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  • 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.

    Mathematics Subject Classification: Primary: 94A55; Secondary: 11T71.

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