# 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:

$C_3 = \pmatrix{0, 0, 0, 0, 0\\0,1,1,0,1\\1,0,1,1,0\\1,1,0,1,1}$

• Tiny notation question: Did your course material typeset it as $Z_2^5$, or $\mathbb{Z}_2^5$? May 27 '21 at 15:40
• It was the former Jun 1 '21 at 5:24

$Z_2^5$ 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:

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