Compressing mappings on primitive sequences over \(\mathbb Z/(2^e)\) and its Galois extension
An element \(a \in \mathbb Z/(2^e)\) can uniquely be written as \(a = a_0+a_12+\cdots+a_{e-1}2^{e-1}\), \(a_i \in \{0,1\}\). Similarly we can decompose a sequence \(\underline{a}=\underline{a}_0+ \underline{a}_12+\cdots+\underline{a}_{e-1}2^{e-1}\) into \(e\) binary sequences \(\underline{a}_i = a_{i0},a_{i1},\ldots\), \(0 \leq i \leq e-1\). If \(\varphi\) is a Boolean function of \(e\) variables we can define a compressing mapping \(\Phi\) from the set of the sequences with terms in \(\mathbb Z/(2^e)\) into the set of binary sequences \(\mathbb F_2^\infty\) by \(\Phi(\underline{a}) = \varphi(\underline{a}_0,\underline{a}_1,\ldots,\underline{a}_{e-1})\), where \(\varphi\) is performed termwise, i.e. the \(j\)th term of the resulting binary sequence equals \(\varphi(a_{0j},a_{1j},\ldots,a_{(e-1)j})\). Let \(f(x)\) be a primitive polynomial of degree \(n\) over \(\mathbb Z/(2^e)\). Then for \(i = 1,2,\ldots, e-1\), we have \(x^{2^{i-1}(2^n-1)}-1 \equiv 2^ih_i(x)\bmod f(x)\), \(h_i(x) \in \mathbb Z/(2^e)[x]\) is of degree less than \(n\) and \(h_i(x)\not\equiv 0\bmod 2\). If \(e \geq 3\) and \(h_2(x)\not\equiv 1\bmod 2\) or \(e = 2\) and \(h_1(x) \not\equiv 1 \bmod 2\), then \(f(x)\) is called a strongly primitive polynomial over \(\mathbb Z/(2^e)\). Suppose that \(\eta\) is a Boolean function of \(e-1\) variables, and let \(G(f(x),\mathbb Z/(2^e))\) denote the set of all sequences with terms in \(\mathbb{Z}/(2^e)\) generated by \(f(x)\). The main result of the first part of the paper is that if \(f(x)\) is a strongly primitive polynomial over \(\mathbb Z/(2^e)\), then the compressing mapping \[ \Phi: \begin{cases} G(f(x),\mathbb Z/(2^e)) \rightarrow \mathbb F_2^\infty,\\ \underline{a}=\underline{a}_0+ \underline{a}_12+\cdots+\underline{a}_{e-1}2^{e-1} \mapsto \varphi(\underline{a}_0, \underline{a}_1,\ldots,\underline{a}_{e-1})\bmod 2,\end{cases} \] where \(\varphi(x_0,x_1,\ldots,x_{e-1}) = x_{e-1}+\eta(x_0,x_1,\ldots,x_{e-2})\) is injective. In the second part of the paper this result is generalized to sequences over the Galois ring \(\text{GR}(2^e,r)\).
scientific article; zbMATH DE number 1891468