Parity-check code


Also known as a sum-zero or even-weight code. An \([n,n-1,2]\) linear binary code whose codewords consist of the message string appended with a parity-check bit such that the parity (i.e., sum over all coordinates of each codeword) is zero. If the Hamming weight of a message is odd (even), then the parity bit is one (zero). This code requires only one extra bit of overhead and is therefore inexpensive.


This code cannot protect information, it can only detect 1-bit error.


The code rate is \(\frac{n}{n+1}\to 1\) as \(n\to\infty\).


If the receiver finds that the parity information of a codeword disagrees with the parity bit, then the receiver will discard the information and request a resend.


Can be realized on almost every communication device. Parity-check codes are some of the earlier error-correcting codes ([1], Ch. 27).




Encyclopedia of Computer Science and Technology, Second Edition Volume I (CRC Press, 2017). DOI
T. R. Oenning and Jaekyun Moon, “A low-density generator matrix interpretation of parallel concatenated single bit parity codes”, IEEE Transactions on Magnetics 37, 737 (2001). DOI
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“Parity-check code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_parity_check, title={Parity-check code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Parity-check code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.