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 $\sum{i=0}^kpi\cdot2^i$. 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 $$a(t)=a0(t)+a1(t)\cdot 2+\cdots+a{k}(t)\cdot2^k,$$ with $ai(t)\in{0, 1}$. Let $\underline{ai}$ denote the binary sequence $(ai(t)){t\ge 0}$ for $0\le i\le k$. Assume $i0$ 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 $i\neq i0$ and $0\le i\le k$, $\underline{ai}=\underline{bi}$ if and only if $\underline{a}=\underline{b}$, and for $i=i0$, $\underline{a{i0}}=\underline{b{i0}}$ if and only if $\underline{a}=\underline{b}$ or $\underline{a}=-\underline{b}$. Then the period of $\underline{ai}$ is equal to the period of $\underline{a}$ if $i\ne i0$ and half of the period of $\underline{a}$ if $i=i0$. We also discuss a possible application of the binary sequences $\underline{ai}$.