On the crosscorrelation of sequences over GF(p) with short periods

We investigate the crosscorrelation function Cd(t)=/spl Sigma//sub i=1/(p/sup n-1/) /spl zeta/(a/sup i-t/-a/sup di/), where /spl zeta/ is a complex primitive pth root of unity, (a/sub i/)(i/spl isin/N/sub 0/) is a maximal linear shift-register sequence of length p/sup n/-1, and p is an odd prime. For p=3, n odd, and d=p/sup n/+1/4+p/sup n/-1/2 we show that 2/spl middot//spl radic/p/sup n/ is an upper bound for the absolute value of 1+C/sub d/(t). For any odd prime p and p/sup k/+1, where n/gcd/(n,k) is not divisible by 4 we determine the maximum absolute value of C/sub d/(t) and the number of values of C/sub d/(t).