Period-Different $m$ -Sequences With at Most Four-Valued Cross Correlation

This paper follows the recent work of Helleseth, Kholosha, Johansen, and Ness to study the cross correlation between an m -sequence of period 2^m - 1 and the d-decimation of an m-sequence of a shorter period 2ⁿ - 1 for an even number m = 2n. Assuming that d satisfies d(2^l + 1) = 2ⁱ (mod 2ⁿ - 1) for some l > 0 and i > 0, it is proved that the cross correlation takes on either exactly three or four values depending on whether I and n are coprime or not. The distribution of the cross-correlation values is also completely determined. Our results theoretically confirm the numerical data by Ness and Helleseth. It is conjectured that there are no other decimations that give at most four-valued cross correlation apart from the ones proved here.