  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.
