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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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