What does this notation stand for when describing a code? This code has appeared in some online course material. 

I understand the $(5, 4, 3)$ refers to (length, num codewords, distance) but no explanation of the $Z_2^5$ notation is given:


  One $(5,4,3)$ code in $Z_2^5$ is given as below:
  
  $C_3 = \pmatrix{0, 0,  0,  0,  0\\0,1,1,0,1\\1,0,1,1,0\\1,1,0,1,1}$
$Z_2^5$ 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:

$Z_2 \times Z_2 = Z_2^2\\
Z_2 \times Z_2 \times Z_2 \times Z_2 \times Z_2 = Z_2^5$