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\(q\)-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.

Cousin

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Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Evaluation code
Evaluation AG codeAG ECC
Generalized RS (GRS) codeMDS Linear \(q\)-ary OA \(t\)-design Universally optimal LRC Distributed-storage ECC
Reed-Solomon (RS) codeAG Evaluation 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 coordinate sums, the cyclic permutation of an SPC codeword is another codeword. The generator polynomial of the code is \(x-1\).
Maximum distance separable (MDS) codeLinear \(q\)-ary OA LRC Distributed-storage \(t\)-design Universally optimal ECC
Small-distance block codeECC
\(q\)-ary parity-check code
Children
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]
Rudolf Schürer and Wolfgang Ch. Schmid. “Extended Reed–Solomon Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CReedSolomon-extended.html
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Zoo Code ID: q-ary_parity_check

Cite as:
“\(q\)-ary parity-check code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_parity_check
BibTeX:
@incollection{eczoo_q-ary_parity_check, title={\(q\)-ary parity-check code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, 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:

“\(q\)-ary parity-check code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_parity_check

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