Period-different m-sequences with at most four-valued cross correlation

Authors:
Tor Helleseth;Lei Hu;Alexander Kholosha;Xiangyong Zeng;Nian Li;Wenfeng Jiang
Affiliations:
Department of Informatics, University of Bergen, Bergen, Norway;Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing, China and State Key Laboratory of Information Security, Graduate University o ...;Department of Informatics, University of Bergen, Bergen, Norway;Faculty of Mathematics and Computer Science, Hubei University, Wuhan, China;Faculty of Mathematics and Computer Science, Hubei University, Wuhan, China;State Key Laboratory of Information Security, Graduate University of Chinese Academy of Sciences, Beijing, China
Venue:
IEEE Transactions on Information Theory
Year:
2009

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Abstract

This paper follows the recent work of Helleseth, Kholosha, Johansen, and Ness to study the cross correlation between an m-sequence of period 2m - 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(2l+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 l 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.