$Z_2^5$ 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 (or the size of the vectors). Here you are using a space of dimension 5, hence 5 coordinates.
$(0,1) \in Z_2^2\\
(0,1,0) \in Z_2^3\\
(0,1,0,1,1) \in Z_2^5$
The exponent notation represent the Cartesian product of the spaces:
$Z_2 \times Z_2 = Z_2^2\\
Z_2 \times Z_2 \times Z_2 \times Z_2 \times Z_2 = Z_2^5$