On the Cross-Correlation of a $p$ -Ary ${m}$ -Sequence of Period

In this paper, for an odd prime , we investigate into the cross-correlation of a p-ary m-sequence m(t) of period p<;sup>;n<;/sup>;-1 and its d-decimated sequences m(dt+l), 0≤l<;(p<;sup>;m<;/sup>;+1)/2, where d=(p^m+1)²/2(p+l), n=2m, and m is an odd integer. There are (p^m+1)/2 distinct decimated sequences m(dt+l) since gcd(d,pⁿ-1)=(p^m+1)/2. It is shown that the magnitude of the cross-correlation values is upper bounded by (p+1)/2 p^n/2+1 . We also construct the sequence family F from these sequences, where the family size is p^m and the correlation magnitude is upper bounded by (p^m+1)/2 p^n/2+1.