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\([n,n-1,2]_q\) \(q\)-ary parity-check code

Alternative Names: \(q\)-ary sum-zero code, \(q\)-ary zero-sum code.

Description

An \([n,n-1,2]_q\) linear \(q\)-ary code whose codewords consist of the message string appended with a parity-check or zero-sum check digit such that the sum over all coordinates of each codeword is zero.

Cousins

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Evaluation codeAG ECC
Evaluation AG code
Residue AG code
Generalized RS (GRS) codeMDS LRC Distributed-storage Linear \(q\)-ary OA Universally optimal ECC \(t\)-design
Reed-Solomon (RS) codeEvaluation AG Linear \(q\)-ary LRC Distributed-storage ECC
Parents
Reed-Solomon (RS) code
RS codes for \(k=n-1\) are parity-check codes [2].
Cyclic linear \(q\)-ary codeLinear \(q\)-ary Cyclic Constacyclic Skew-cyclic ECC
Since permutations preserve coordinate sums, the cyclic permutation of an SPC codeword is another codeword. The generator polynomial of the code is \(x-1\).
Checksum codeECC
The \(q\)-ary parity check code is a simple example of a checksum code, with the parity of the message being the checksum.
Small-distance block codeECC
\([n,n-1,2]_q\) \(q\)-ary parity-check code
Children
\([n,n-1,2]\) Single parity-check (SPC) code

References

[1]
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
[2]
R. Schürer and W. Ch. Schmid, “Extended Reed–Solomon Code”, From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03 URL
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Zoo Code ID: q-ary_parity_check

Cite as:
“\([n,n-1,2]_q\) \(q\)-ary parity-check code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/q-ary_parity_check, arXiv:2606.11484
BibTeX:
@incollection{eczoo_q-ary_parity_check,
title={\([n,n-1,2]_q\) \(q\)-ary parity-check code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/q-ary_parity_check}
}
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Permanent link:
https://errorcorrectionzoo.org/c/q-ary_parity_check

Cite as:

“\([n,n-1,2]_q\) \(q\)-ary parity-check code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/q-ary_parity_check, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/easy/q-ary_parity_check.yml.