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$\mathbf{Z}_2^b$ is the direct product of $b$ copies of $\mathbf{Z}_2$ ($\mathbf{Z}_2 \times\cdots \times \mathbf{Z}_2$, $b$ times). That is, its elements are $b$-tuples of elements of $\mathbf{Z}_2$, with both addition and multiplication defined componentwise. If $b > 1$, it is not a field. In fact it is not even an integral domain, because $(0,1)\times ...


1

We can present $\mathbb{Z}_m$ in different manner. For example, bellow sets are some equivalence classes of $\mathbb{Z}_3$ $$\{0,1,2\}, \{3,4,5\},\{-3,-2,-1\}. $$ Now, about your question If you select the numbers of $[-n,n]$ module a positive integer $m$ which $m\leq 2n+1$, you can construct $\mathbb{Z}_m$: $$\mathbb{Z}_m=\{i \pmod m \mid i\in[-n,n]\}$$ ...



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