The Unique Distribution of Zeros in CompressingSequences Derived from Primitive Sequencesover Z=(pe)
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Graphical Abstract
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Abstract
Let Z=(pe) be the integer residue ring with
odd prime p and integer e ¸ 3. Any sequence a over Z=(pe)
has a unique p-adic expansion a = a0+a1 ¢p+¢ ¢ ¢+ae¡1 ¢pe¡1,
where ai can be regarded as a sequence over Z=(p) for
0 · i · e ¡ 1. Let f(x) be a strongly primitive polynomial
over Z=(pe) and let a; b be two primitive sequences gener-
ated by f(x) over Z=(pe). Assume '(x0; ¢ ¢ ¢ ; xe¡1) = xe¡1 +
´(x0; ¢ ¢ ¢ ; xe¡2), where the degree of xe¡2 in ´(x0; ¢ ¢ ¢ ; xe¡2)
is less than p¡1. It is shown that if '(a0(t); ¢ ¢ ¢ ; ae¡1(t)) = 0
if and only if '(b0(t); ¢ ¢ ¢ ; be¡1(t)) = 0 for all nonnegative
integer t with ®(t) 6= 0, where ®, is an m-sequence de-
termined by f(x) and a0, then a = b. In particular, when
´(x0; ¢ ¢ ¢ ; xe¡2) = 0, it is just the former result on the unique
distribution of zeros in the highest level sequences.
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