Finding the cross correlation between two m-sequences {st} and {sdt} of the same period 2m-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 = 22k+1/2k+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.

Recently, the complete five-valued cross-correlation distribution has been determined between twom-sequences{st} and{sdt} of periods2m-1 that differ by the decimationd= [(22k+1)/(2k+1)] wheremis odd andk= 1, i.e.,d= 5/3 . In this paper, the correlation distribution is given in terms of some exponential sums for anykwhengcd(k,m) = 1 andmis 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 ofkunder the conditions above and thus the same as for the casek= 1. The conjectures are proven formodd andk= 2 and the paper gives a new result that the correlation distribution ford=17/5 is the same as ford= 5/3.

Considered is the distribution of the cross correlation between m-sequences of length 2m-1, where m is even, and m-sequences of shorter length 2m/2-1. The infinite family of pairs of m-sequences with four-valued cross correlation is constructed and the complete correlation distribution of this family is determined.

Family A is a family of sequences of period 2n- 1 over Zi, the ring of integers modulo 4. This family has optimal correlation properties and its correlation distribution is well known. Two related families of quaternary sequences are the families B and C. These are families of sequences over Z4of period 2(2n- 1). In recent years, new families of quaternary sequences of period 2(2n- 1) have been constructed by modifying the sequence families B and C in a nonlinear way. This has resulted in a new family D of sequences of period 2(2n- 1) which has optimal correlation properties, but until now the correlation distribution of this family has not been known. In this paper, we completely determine the correlation distribution of family D by making use of properties of exponential sums.