A New Family Of Binary Sequences With Four-valued Cross Correlations Of Different Period

Sequences with low cross correlation are special class of sequences with desirable algebraic properties. They have been widely used in radar, CDMA communication system and other fields. Sequences with low cross correlation of same period have been studied nearly 40 years. Recently, there has been intensive research on the correlation distribution of m-sequences. They have gotten a lot of good results. Especially in the 1970s, Trachtenberg, Niho and Helleseth at all published a lot of influential thesis on this topic. Sequences with low cross correlation of different period are an important part of the sequences' cross correlation distribution. In this paper,let k be odd and q = 22k. We study the cross-correlation between a binary m-sequence u(t) with period q-1 and vl(t) = u(dt + l) with period (q-1)/3, 0 ?l?2. d and s respectively satisfying d = (2k-1)s + 1 and (22r + 1)s = 1( mod2k + 1) . Combining the theory of quadratic form over finite field and some skills on solving algebraic equations. We completely determine the correlations distributions. The correlation is shown to be four values.When l= 0, we can get the values of the correlation satisfying the above con-ditions. There are -1, -1 -2k, -1+ 2k, -1 + 2k+1. They respectively occurs(22k-1) - 2k-1 +2, (2k+1)(2k-2)/3,2k-2,(2k+1)(2k-1-1)/3.When l= 1 or 2, we can get the same values of the correlation satisfying the above conditions. There are -1, -1 -2k, -1 + 2k, -1 + 2k+1. They respectively occurs (22k - 1) - 2k- 1, (2k+1)(2k-2)/3,2k + 1, (2k+1)(k-1+1)/3.