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Since [7, 4, 3] = [''n'', ''k'', ''d''] = [2<sup>''m''</sup> − 1, 2<sup>''m''</sup>−1−''m'', 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''
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
The code [[generator matrix]] <math>\mathbf{G}</math> and the [[parity-check matrix]] <math>\mathbf{H}</math> are:
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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>
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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>
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