Distribution Properties of Compressing Sequences Derived From Primitive Sequences Modulo Odd Prime Powers

Let \(\underline {a}\) and \(\underline {b}\) be primitive sequences over \(\mathbb {Z}/(p^{e})\) with odd prime \(p\) and \(e\ge 2\) . For certain compressing maps, we consider the distribution properties of compressing sequences of \(\underline {a}\) and \(\underline {b}\) , and prove that \(\underline {a}=\underline {b}\) if the compressing sequences are equal at the times \(t\) such that \(\alpha (t)=k\) , where \(\underline {\alpha }\) is a sequence related to \(\underline {a}\) . We also discuss the \(s\) -uniform distribution property of compressing sequences. For some compressing maps, we obtain that there exist different primitive sequences such that the compressing sequences are \(s\) -uniform. We also discuss that for how many elements \(s\) , compressing sequences of different primitive sequences can be \(s\) -uniform.