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11

$Z_2^5$ means that you are working in $GF(2)^5$. $GF(2)$ is the Finite Field with two elements: 0 and 1 with the addition and multiplications defined: $0 + 0 = 0\\ 0 + 1 = 1\\ 1 + 0 = 1\\ 1 + 1 = 0$ It is equivalent to XOR. $0 \times 0 = 0\\ 0 \times 1 = 0\\ 1 \times 0 = 0\\ 1 \times 1 = 1$ It is equivalent to AND. the $ ^5$ is the dimension of the space ...


5

$\mathbf{Z}_2^b$ 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 (...



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