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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
specifies that for each codeword, digits 1 and 2 should sum to zero (according to the second row) and digits 3 and 4 should sum to zero (according to the first row).
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 is in standard form
then the parity check matrix is given by
Negation is performed in the finite field Fq. Note that if the characteristic of the underlying field is 2 (i.e., 1 + 1 = 0 in that field), as in binary codes, then -P = P, so the negation is unnecessary.
For example, if a binary code has the generator matrix
then its parity check matrix is
For any (row) vector x of the ambient vector space, s = Hxt is called the syndrome of x. The vector x is a codeword if and only if s = 0.
- Hill, Raymond (1986). A first course in coding theory. Oxford Applied Mathematics and Computing Science Series. Oxford University Press. p. 69. ISBN 0-19-853803-0.
- Pless, Vera (1982). Introduction to the theory of error-correcting codes. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons. p. 8. ISBN 0-471-08684-3.
- J.H. van Lint (1992). Introduction to Coding Theory. GTM 86 (2nd ed ed.). Springer-Verlag. p. 34. ISBN 3-540-54894-7.
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