Some New Results on the Cross Correlation of $m$-sequences

The determination of the cross correlation between an m-sequence and its decimated sequence has been a long-standing research problem. Considering a ternary m-sequence of period 3 − 1, we determine the cross correlation distribution for decimations d = 3 + 2 and d = 3 + 2, where gcd(r, 3) = 1. Meanwhile, for a binary m-sequence of period 2 − 1, we make an initial investigation for the decimation d = 2 2lm −1 2m+1 +2, 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. Index Terms cross correlation, cross correlation distribution, decimation, m-sequences, Weil sums