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  

Publication
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
Keyword: 
shift-and-add property,  m-sequences,  trinomial,  reciprocal polynomial,  cyclotomic cosets,  

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Summary: 
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.