Characterization of $m$-Sequences of Lengths $2^{2k}-1$ and $2^{k}-1$ With Three-Valued Cross Correlation

Considered is the distribution of the cross correlation between in-sequences of length 2
^2k
-1, where m = 2k, and m-sequences of shorter length 2
^k
-1. New pairs of m -sequences with three-valued cross correlation are found and the complete correlation distribution is determined. Finally, we conjecture that there are no more cases with a three-valued cross correlation apart from the ones proven here.