A note on the five valued conjectures of Johansen and Helleseth and zeta functions

Abstract: 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 $d={{2^{2k}+1}\over {2^k+1}}$ where $m$ is odd and $\mbox{gcd}(k,m)=1$, Johansen and Hellseth expressed it in terms of some exponential sums. And two conjectures are presented that are of interest in their own right. 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. Here the multiplicity of the biggest absolute value of the cross-correlation is no more than one-sixth of the multiplicity corresponding the smallest absolute value.