Single parity-check (SPC) code 

Also known as Sum-zero code, Zero-sum code, Even-weight code.


An \([n,n-1,2]\) linear binary code whose codewords consist of the message string appended with a parity-check bit or parity 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. Its codewords are all even-weight binary strings. Its automorphism group is \(S_n\).


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.Wagner's rule yields a procedure that is linear in \(n\) [1] (see [2; Sec. 29.7.2] for a description).


Can be realized on almost every communication device. SPCs are some of the earliest error-correcting codes ([3], Ch. 27).




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P. Boyvalenkov, D. Danev, "Linear programming bounds." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) 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
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
N. Rengaswamy et al., “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) arXiv:1803.06987 DOI
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Zoo Code ID: parity_check

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“Single parity-check (SPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_parity_check, title={Single parity-check (SPC) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Single parity-check (SPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.