Proof of a conjecture of Sarwate and Pursley regarding pairs of binary m-sequences

Binary m-sequences are maximal length sequences generated by shift registers of length m, that are employed in navigation, radar, and spread-spectrum communications systems, because of their crosscorrelation properties. It is well known that given a pair of distinct m-sequences, the crosscorrelation function must take on at least three values. The article considers crosscorrelation functions that take on exactly three values, and where these values are preferred in that they are small. The main result is a proof of a conjecture made by Sarwate and Pursley in 1980, that if m≡0 (mod 4) then there are no preferred pairs of binary m-sequences. The proof makes essential use of a deep theorem of McEliece (1971) that restricts the possible weights that can occur in a binary cyclic code