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\(q\)-ary repetition code

Description

An \([n,1,n]_q\) code encoding consisting of codewords \((j,j,\cdots,j)\) for \(j \in \mathbb{F}_q\).

Protection

Detects errors on up to \(\frac{n-1}{2}\) coordinates, corrects erasure errors on up to \(\frac{n-1}{2}\) coordinates.

Decoding

The receiver can use majority vote to recover the information.

Cousin

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Evaluation code
Polynomial evaluation code
Generalized RM (GRM) codeLinear \(q\)-ary LCC LRC Distributed-storage LDC ECC
Parents
Generalized RM (GRM) code
GRM\(_q(0,m)\) codes are evaluations of all zero-degree polynomials on \(\mathbb{F}_q^n\), which are just the \(q\) constant polynomials. Therefore, \(q\)-ary repetition codes are GRM\(_q(0,m)\) codes.
Reed-Solomon (RS) codeGRS Evaluation MDS Linear \(q\)-ary OA AG Universally optimal LRC Distributed-storage ECC \(t\)-design
\(q\)-ary repetition codes can be thought of as RS codes [2].
Extended GRS codeGRM Evaluation Linear \(q\)-ary LCC LRC Distributed-storage LDC ECC
\(q\)-ary repetition codes can be thought of as extended RS codes [2].
Cyclic linear \(q\)-ary codeCyclic Linear \(q\)-ary Constacyclic Skew-cyclic ECC
The \(q\)-ary repetition code is cyclic with generator polynomial \(1+x+\cdots+x^{n-1}\).
\(q\)-ary sharp configurationOA Sharp configuration Universally optimal ECC \(t\)-design
The \(q\)-ary repetition code is a \(q\)-ary sharp configuration [3; Table 12.1].
\(q\)-ary repetition code
Children
Repetition code
\(q\)-ary repetition code reduce to repetition codes for \(q=2\).

References

[1]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[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://web.archive.org/web/20240420202309/https://mint.sbg.ac.at/desc_CReedSolomon-extended.html
[3]
P. Boyvalenkov, D. Danev, “Linear programming bounds.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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Zoo Code ID: q-ary_repetition

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

Cite as:

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

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