Description
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, and its parity-check matrix is a row vector of all ones. Its automorphism group is \(S_n\).Protection
This code detects any odd number of bit errors but cannot correct an arbitrary single-bit error.Rate
The code rate is \(\frac{n-1}{n}\to 1\) as \(n\to\infty\).Decoding
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).Realizations
Can be realized on almost every communication device. SPCs are some of the earliest error-correcting codes [3; Ch. 27].Cousins
- Reed-Muller (RM) code— Binary SPC codes of length \(2^m\) are RM\((m-1,m)\) codes.
- Repetition code— Binary SPCs and repetition codes are dual to each other.
- Dual linear code— Binary SPCs and repetition codes are dual to each other.
- Linear binary code— Any \([n,k,d]\) code with odd distance can be extended to an \([n+1,k,d+1]\) code by adding a bit storing the sum of codeword coordinates.
- Low-density generator-matrix (LDGM) code— Concatenated SPCs are LDGM [6].
- \(D_n\) checkerboard lattice— \([n,n-1,2]\) SPC codes yield \(D_n\) checkerboard lattices via Construction A [7; Exam. 10.5.1][8; pg. 138].
- Klemm code— The generator matrix of the Klemm code consists of a sum of the generator matrix of the repetition code and twice the generator matrix of the SPC code [9].
- \([[2m,2m-2,2]]\) error-detecting code— The \([[2m,2m-2,2]]\) error-detecting code is constructed via the CSS construction from an SPC code and its dual repetition code [10; Sec. III].
- \([[7,3,2]]\) punctured hypercube code— The \([[7,3,2]]\) punctured hypercube code \(H_Z\) check matrix is the parity-check matrix of the \([7,4,3]\) Hamming code, while its \(H_X\) matrix is that of the SPC code.
- \([[8,3,2]]\) Smallest interesting color code— The \([[8,3,2]]\) hypercube code \(H_Z\) check matrix is the parity-check matrix of the \([8,4,4]\) extended Hamming code, while its \(H_X\) matrix is that of the SPC code.
- Classical-product code— SPC codes are used as component codes in classical-product code constructions.
Member of code lists
- Algebraic-geometry codes
- Binary linear codes
- Classical codes
- Classical codes with a rate
- Classical codes with notable decoders
- Cyclic codes
- Evaluation codes
- Locally recoverable codes
- MDS codes and generalizations
- Orthogonal arrays and friends
- Realized classical codes
- Reed-Solomon codes and friends
- Small-distance classical codes and friends
- Universally optimal codes
Primary Hierarchy
References
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R. Silverman and M. Balser, “Coding for constant-data-rate systems”, Transactions of the IRE Professional Group on Information Theory 4, 50 (1954) DOI
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A. Lapidoth, A Foundation in Digital Communication (Cambridge University Press, 2017) DOI
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Encyclopedia of Computer Science and Technology, Second Edition Volume I (CRC Press, 2017) DOI
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W. W. Peterson, “Error-Correcting Codes”, Scientific American 206(2), 96–111 (1962) URL
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J. H. Conway and R. K. Guy, The Book of Numbers (Springer New York, 1996) DOI
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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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T. Ericson and V. A. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
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J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
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S. T. Dougherty, P. Gaborit, M. Harada, A. Munemasa, and P. Sole, “Type IV self-dual codes over rings”, IEEE Transactions on Information Theory 45, 2345 (1999) DOI
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N. Rengaswamy, R. Calderbank, H. D. Pfister, and S. Kadhe, “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) 791 (2018) arXiv:1803.06987 DOI
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M. Ergen, “Basics of Cellular Communication”, Mobile Broadband 19 (2008) DOI
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J. Conway and N. Sloane, “Lexicographic codes: Error-correcting codes from game theory”, IEEE Transactions on Information Theory 32, 337 (1986) DOI
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P. Boyvalenkov, D. Danev, “Linear programming bounds”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 251-266 DOI
Page edit log
- Victor V. Albert (2026-10-13) — most recent
- Victor V. Albert (2026-06-08)
- Victor V. Albert (2022-11-08)
- Victor V. Albert (2022-07-20)
- Victor V. Albert (2021-12-15)
- Yijia Xu (2021-12-14)
Cite as:
“\([n,n-1,2]\) Single parity-check (SPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/parity_check, arXiv:2606.11484