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In coding theory, a parity-check matrix of a linear block code C
is a generator matrix of the dual code. As such, a codeword c is in C if and only if the matrix-vector product H^{T}c=0.## Creating a parity check matrix

The parity check matrix for a given code can be derived from its generator matrix (and vice-versa). If the generator matrix for an [n,k]-code in standard form is
## References

The rows of a parity check matrix are parity checks on the codewords of a code. That is, they show how linear combinations of certain digits of each codeword equal zero. For example, the parity check matrix

$H\; =$

begin{bmatrix}

0011

1100end{bmatrix}

specifies that for each codeword, digits 1 and 2 should sum to zero and digits 3 and 4 should sum to zero.

For more information see Hamming code and generator matrix.

- $G\; =\; begin\{bmatrix\}\; I\_k\; |\; P\; end\{bmatrix\}$

the parity check matrix can be calculated as

- $H\; =\; begin\{bmatrix\}\; -P^T\; |\; I\_\{n-k\}\; end\{bmatrix\}$

For example, if a binary code has the generator matrix

- $G\; =$

The parity check matrix becomes

- $H\; =$

For any valid codeword $x$, $Hx\; =\; 0$. For any invalid codeword $tilde\{x\}$, the syndrome $S$ satisfies $Htilde\{x\}\; =\; S$.

- Hill, Raymond
*A first course in coding theory*. Oxford University Press. - Pless, Vera
*Introduction to the theory of error-correcting codes*. John Wiley & Sons. - J.H. van Lint
*Introduction to Coding Theory*. 2nd ed, Springer-Verlag.

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Last updated on Friday July 11, 2008 at 14:37:47 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday July 11, 2008 at 14:37:47 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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