A Family of $m$ -Sequences With Five-Valued Cross Correlation

Finding the cross correlation between two m-sequences {s_t} and {s_dt} of the same period 2^m -1 , that differ by a decimation d, has been a popular research problem since the 1960s. Many cases with three- and four-valued correlation have been determined. Several values of d are known to lead to five-valued cross correlation but their exact correlation distribution has been open. The correlation distribution is completely determined for one of these families with five-valued cross correlation by using evaluations of certain exponential sums including Kloosterman sums. The decimation considered is the special decimation d = 2^2k+1/2^k+1where m is odd and k=1, i.e., d=5/3 . The paper introduces some techniques that may be useful to obtain further results on related decimations.