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Some new three-valued crosscorrelation functions for binary $$m$$-sequences. (English) Zbl 0855.94012
The crosscorrelation between two binary maximal-length sequences $$a_t$$ and $$a_{dt}$$ of period $$n= 2^n-1$$ is defined as $$C_d (t)= \sum^{2^n- 2}_{j= 0} (-1)^{a_{jd}+ a_{j+t}}$$ for $$t= 0, 1, \dots, 2^n- 2$$. The authors consider $$C_d (t)$$ for the following two new values of $$d$$, $$d= 2^m+ 2^{(m+1) /2}+ 1$$ and $$d= 2^{m+1}+ 3$$ where $$n= 2m$$ and $$m$$ odd. They determine the spectrum of the crosscorrelation function $$C_d (t)$$ and show that it takes on only three different values $$-1$$ and $$-1\pm 2^{m+1}$$ and thereby prove two conjectures due to Niho in his 1972 thesis.

##### MSC:
 94A55 Shift register sequences and sequences over finite alphabets in information and communication theory 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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