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 3^3r-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 $l\geq 2$ is even and $0\leq s\leq 2m-1$ . 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. [ABSTRACT FROM AUTHOR]