Editing Hamming code (section)

===Encoding===
;Example

From the above matrix we have 2<sup>k</sup> = 2<sup>4</sup> = 16 codewords.
Let <math>\vec{a}</math> be a row vector of binary data bits, <math> \vec{a}=[a_1,a_2,a_3,a_4],\quad a_i\in \{0,1\} </math>. The codeword <math>\vec{x}</math> for any of the 16 possible data vectors <math> \vec{a} </math> is given by the standard matrix product <math>\vec{x}=\vec{a}G </math> where the summing operation is done modulo-2.

For example, let <math>\vec{a}=[1,0,1,1]</math>. Using the generator matrix <math>G</math> from above, we have (after applying modulo 2, to the sum),

<math>\vec{x}=\vec{a}G= \begin{pmatrix} 1 & 0 & 1 & 1 \end{pmatrix}\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} = \begin{pmatrix} 1 & 0 & 1 & 1 & 2 & 3 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 1 & 1 & 0 & 1 & 0 \end{pmatrix}</math>