Some New Results on the Cross Correlation of -Sequences

The determination of the cross correlation between an m-sequence and its decimated sequence has been a longstanding research problem. Considering a ternary m-sequence of period 3^3r - 1, we determine the cross correlation distribution for decimations d = 3^r + 2 and d = 3^2r + 2, where gcd(r, 3) = 1. Meanwhile, for a binary m-sequence of period 2^2lm - 1, we make an initial investigation for the decimation d = (2^2lm - 1)/(2^m + 1) + 2^s, where l ≥ 2 is even and 0 <; s <; 2m - 1. It is shown that the cross correlation takes at least four values. Furthermore, we confirm the validity of two famous conjectures due to Sarwate et al. and Helleseth in this case.