The Shift-and-Add Property of m-Sequences

Fanxin ZENG  Lijia GE  Xiping HE  Guixin XUAN  Guojun LI  Zhenyu ZHANG  Yanni PENG  Linjie QIAN  Sheng LU  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E102-A    No.4    pp.685-690
Publication Date: 2019/04/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E102.A.685
Type of Manuscript: LETTER
Category: Information Theory
shift-and-add property,  m-sequences,  trinomial,  reciprocal polynomial,  cyclotomic cosets,  

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The shift-and-add property (SAP) of a p-ary m-sequence {ak} with period N=pn-1 means that this sequence satisfies the equation {ak+η}+{ak+τ}={ak+λ} for some integers η, τ and λ. For an arbitrarily-given p-ary m-sequence {ak}, we develop an algebraic approach to determine the integer λ for the arbitrarily-given integers η and τ. And all trinomials can be given. Our calculation only depends on the reciprocal polynomial of the primitive polynomial which produces the given m-sequence {ak}, and the cyclotomic cosets mod pn-1.