Further Results on $m$ -Sequences With Five-Valued Cross Correlation

Recently, the complete five-valued cross-correlation distribution has been determined between two m-sequences {st} and {sdt} of periods 2 ^m-1 that differ by the decimation d = [(2² k+1)/(2k+1)] where m is odd and k = 1, i.e., d = 5/3 . In this paper, the correlation distribution is given in terms of some exponential sums for any k when gcd(k,m) = 1 and m is odd. Furthermore, two new conjectures on exponential sums are presented that are of interest in their own right. Proving these conjectures would imply that the correlation distribution is independent of k under the conditions above and thus the same as for the case k = 1. The conjectures are proven for m odd and k = 2 and the paper gives a new result that the correlation distribution for d=17/5 is the same as for d = 5/3.