Editing Hamming code (section)

===Construction of G and H===
The matrix 
<math>\mathbf{G} := \begin{pmatrix} \begin{array}{c|c}
I_k & -A^\text{T} \\
\end{array} \end{pmatrix}</math> is called a (canonical) generator matrix of a linear (''n'',''k'') code,

and <math>\mathbf{H} := \begin{pmatrix} \begin{array}{c|c}
A & I_{n-k} \\
\end{array} \end{pmatrix}</math> is called a [[parity-check matrix]].

This is the construction of '''G''' and '''H''' in standard (or systematic) form. Regardless of form, '''G''' and '''H''' for linear block codes must satisfy

<math>\mathbf{H}\,\mathbf{G}^\text{T} = \mathbf{0}</math>, an all-zeros matrix.<ref name=Moon>Moon T. Error correction coding: Mathematical Methods and
Algorithms. John Wiley and Sons, 2005.(Cap. 3) {{ISBN|978-0-471-64800-0}}</ref>

Since [7,&nbsp;4,&nbsp;3] =&nbsp;[''n'',&nbsp;''k'',&nbsp;''d''] =&nbsp;[2<sup>''m''</sup>&nbsp;−&nbsp;1, 2<sup>''m''</sup>&nbsp;−&nbsp;1&nbsp;−&nbsp;''m'',&nbsp;3]. The [[parity-check matrix]] '''H''' of a Hamming code is constructed by listing all columns of length ''m'' that are pair-wise independent.

Thus '''H''' is a matrix whose left side is all of the nonzero ''n''-tuples where order of the ''n''-tuples in the columns of matrix does not matter. The right hand side is just the (''n''&nbsp;−&nbsp;''k'')-[[identity matrix]].

So '''G''' can be obtained from '''H''' by taking the transpose of the left hand side of '''H''' with the identity ''k''-[[identity matrix]] on the left hand side of&nbsp;'''G'''.

The code [[generator matrix]] <math>\mathbf{G}</math> and the [[parity-check matrix]] <math>\mathbf{H}</math> are:

<math>\mathbf{G} := \begin{pmatrix}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1 & 1 & 1 & 1
\end{pmatrix}_{4,7}</math>

and

<math>\mathbf{H} := \begin{pmatrix}
1 & 1 & 0 & 1 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{pmatrix}_{3,7}.</math>

Finally, these matrices can be mutated into equivalent non-systematic codes by the following operations:<ref name=Moon/>

* Column permutations (swapping columns)
* Elementary row operations (replacing a row with a linear combination of rows)