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

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 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.