Uniqueness of the distribution of zeroes of primitive level sequences over Z/(pe) (II)

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Abstract

Let p be a prime number, Z/(pe) the integer residue ring, e2. For a sequence a̲ over Z/(pe), there is a unique decomposition a̲=a̲0+a̲1p++a̲e1pe1, where a̲i be the sequence over {0,1,,p1}. Let f(x)Z/(pe)[x] be a primitive polynomial of degree n, a̲ and b̲ be sequences generated by f(x) over Z/(pe), such that a̲0̲(modpe1). This paper shows that the distribution of zero in the sequence a̲e1=(ae1(t))t0 contains all information of the original sequence a̲, that is, if ae1(t)=0 if and only if be1(t)=0 for all t0, then a̲=b̲. Here we mainly consider the case of p=3 and the techniques used in this paper are very different from those we used for the case of p5 in our paper [X.Y. Zhu, W.F. Qi, Uniqueness of the distribution of zeroes of primitive level sequences over Z/(pe), Finite Fields Appl. 11 (1) (2005) 30–44].

Keywords

Integer residue ring
Linear recurring sequence
Primitive sequence
Level sequence

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This work was supported by the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant 200060) and the National Natural Science Foundation of China (Grant 60373092).