Binary m-sequences with three-valued crosscorrelation: a proof of Welch's conjecture

Abstract

We prove the long-standing conjecture of Welch stating that for odd n=2m+1, the power function x/sup d/ with d=2/sup m/+3 is maximally nonlinear on GF(2/sup n/) or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence of degree n and a decimation of that sequence by 2/sup m/+3 takes on precisely the three values -1, -1/spl plusmn/2/sup m+1/.