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Compression mappings on primitive sequences over Z/(pe)

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2 Author(s)
Xuan Yong Zhu ; Dept. of Appl. Math., Zhengzhou Inf. Eng. Univ., China ; Wen Feng Qi

Let Z/(pe) be the integer residue ring with odd prime p≥5 and integer e≥2. For a sequence a_ over Z/(pe), there is a unique p-adic expansion a_=a_0+a_·p+...+a_e-1·pe-1, where each a_i is a sequence over {0,1,...,p-1}, and can be regarded as a sequence over the finite field GF(p) naturally. Let f(x) be a primitive polynomial over Z/(pe), and G'(f(x),pe) the set of all primitive sequences generated by f(x) over Z/(pe). Set φe-1 (x0,...,xe-1) = xe-1k + ηe-2,1(x0, x1,...,xe-2) ψe-1(x0,...,xe-1) = xe-1k + ηe-2,2(x0,x1,...,xe-2) where ηe-2,1 and ηe-2,2 are arbitrary functions of e-1 variables over GF(p) and 2≤k≤p-1. Then the compression mapping φe-1:{G'(f(x),pe) → GF(p) a_ → φe-1(a_0,...,a_e-1) is injective, that is, a_ = b_ if and only if φe-1(a_0,...,a_e-1) = φe-1(b_0,...,b_e-1) for a_,b_ ∈ G'(f(x),pe). Furthermore, if f(x) is a strongly primitive polynomial over Z/(pe), then φe-1(a_0,...,a_e-1) = ψe-1(b_0,...,b_e-1) if and only if a_ = b_ and φe-1(x0,...,xe-1) = ψe-1(x0,...,xe-1) for a_,b_ ∈ G'(f(x),pe).

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Information Theory, IEEE Transactions on  (Volume:50 ,  Issue: 10 )