On the distinctness of binary sequences derived from $2$-adic expansion of m-sequences over finite prime fields

Let $p$ be an odd prime with $2$-adic expansion $\sum_{i=0}^kp_i\cdot2^i$. For a sequence $\underline{a}=(a(t))_{t\ge 0}$ over $\mathbb{F}_{p}$, each $a(t)$ belongs to $\{0,1,\ldots, p-1\}$ and has a unique $2$-adic expansion $a(t)=a_0(t)+a_1(t)\cdot 2+\cdots+a_{k}(t)\cdot2^k,$ with $a_i(t)\in\{0, 1\}$. Let $\underline{a_i}$ denote the binary sequence $(a_i(t))_{t\ge 0}$ for $0\le i\le k$. Assume $i_0$ is the smallest index $i$ such that $p_{i}=0$ and $\underline{a}$ and $\underline{b}$ are two different m-sequences generated by a same primitive characteristic polynomial over $\mathbb{F}_p$. We prove that for $i\neq i_0$ and $0\le i\le k$, $\underline{a_i}=\underline{b_i}$ if and only if $\underline{a}=\underline{b}$, and for $i=i_0$, $\underline{a_{i_0}}=\underline{b_{i_0}}$ if and only if $\underline{a}=\underline{b}$ or $\underline{a}=-\underline{b}$. Then the period of $\underline{a_i}$ is equal to the period of $\underline{a}$ if $i\ne i_0$ and half of the period of $\underline{a}$ if $i=i_0$. We also discuss a possible application of the binary sequences $\underline{a_i}$.

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