Cross correlation of m-sequences-some unusual coincidences

In general, the problem of computing cross-correlations between m-sequences by analytical techniques has been declared untractable. In this research, cross-correlations for all sequences of lengths up to 2 ¹⁸-1 have been examined numerically, in the hope of finding some predictable patterns. One pattern which emerged from this numerical analysis was the existence of cross-correlation peaks well in excess of those predicted by statistical techniques. This paper demonstrates these “anomalous” peaks to be due to finite algebra effects. These results suggest the existence of some algebraically computable correlations, apart from those already known from Galois field theory. The technique developed here can be used to determine the pairs of sequences with high cross-correlation peaks, the approximate value of these peaks and the relative phasing of the sequences. Elimination of such pairs of sequences results in a dramatic reduction in the peak cross-correlation among the set of remaining sequences. These have been named “constrained connected sets”. When used in conjunction with sequences whose cross-correlations are predictable by Galois field analysis, this technique may prove useful in the design of code families to meet specific cross-correlation requirements