On the Cross-Correlation of a $p$-Ary ${m}$-Sequence of Period $p^{2m}-1$ and Its Decimated Sequences by $ {{ (p^{m}+1)^{2}}/ { 2(p+1)}}$

Abstract

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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> +1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /2(p+l), n=2m, and m is an odd integer. There are (p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> +1)/2 distinct decimated sequences m(dt+l) since gcd(d,p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -1)=(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> +1)/2. It is shown that the magnitude of the cross-correlation values is upper bounded by (p+1)/2 p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n/2</sup> +1 . We also construct the sequence family F from these sequences, where the family size is p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> and the correlation magnitude is upper bounded by (p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> +1)/2 p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n/2</sup> +1.