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 $d=\frac{2^{2lm}-1}{2^{m}+1}+2^{s}$, where $l \ge 2$ is even and $0 \le s \le 2m-1$. 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.