有限素域上m序列的$2$ -adic展开式所导出的二元序列的区别性
On the distinctness of binary sequences derived from $2$-adic expansion of m-sequences over finite prime fields
摘要：
令$p$是一个奇素数，具有$2$ -adic扩展$\sum_{i=0}^kp_i\cdot2^i$。对于$\mathbb{F}_{p}$上的序列$\underline{a}=(a(t))_{t\ge 0}$，每个$a(t)$属于$\{0,1,\ldots, p-1\}$，并且具有唯一的$2$ -进扩展$$a(t)=a_0(t)+a_1(t)\cdot 2+\cdots+a_{k}(t)\cdot2^k,$$和$a_i(t)\in\{0, 1\}$。令$\underline{a_i}$表示$0\le i\le k$的二进制序列$(a_i(t))_{t\ge 0}$。假设$i_0$是最小索引$i$，使得$p_{i}=0$和$\underline{a}$和$\underline{b}$是由$\mathbb{F}_p$上的同一本原特征多项式生成的两个不同的m序列。证明了对于$i\neq i_0$和$0\le i\le k$，$\underline{a_i}=\underline{b_i}$当且仅当$\underline{a}=\underline{b}$，对于$i=i_0$，$\underline{a_{i_0}}=\underline{b_{i_0}}$当且仅当$\underline{a}=\underline{b}$或$\underline{a}=-\underline{b}$ .则$\underline{a_i}$的周期在$i\ne i_0$的情况下等于$\underline{a}$的周期，在$i=i_0$的情况下等于$\underline{a}$的周期的一半。我们还讨论了二进制序列$\underline{a_i}$的一个可能的应用。
英文摘要：
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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