A Note on the Five Valued Conjectures of Johansen, Helleseth and Kholosha and Zeta Functions

摘要

For the complete five-valued cross-correlation distribution between two $m$-sequences $s_{t}$ and $s_{dt}$ of period $2^{m}-1$ that differ by the decimation $d=(2^{2k}+1)/(2^{k}+1)$ where $m$ is odd and $hbox{gcd}(k,m)=1$, Johansen and Helleseth expressed it in terms of some exponential sums. And two conjectures were presented about them. In this correspondence we study these conjectures for the particular case where $k=3$, and the cases $k=1,2$ can also be analyzed in a similar process. When $k>3$, the degrees of the relevant polynomials will become higher.