Description
An \([n,1,n]_q\) code consisting of codewords \((j,j,\cdots,j)\) for \(j \in \mathbb{F}_q\).Protection
Detects up to \(n-1\) symbol errors, corrects up to \(\left\lfloor (n-1)/2\right\rfloor\) symbol errors by majority vote, and corrects up to \(n-1\) erasures.Rate
Code rate is \(1/n\), and code distance is \(n\).Decoding
The receiver can use majority vote to recover the information.Cousins
- Locally recoverable code (LRC)— The \(q\)-ary repetition code is an LRC with \(r=2\) [1].
- \(\Lambda_{16}\) Barnes-Wall lattice— The \(\Lambda_{16}\) Barnes-Wall lattice can be obtained from the \([4,1,4]\) repetition code over \(\mathbb{F}_9\) via Quebbemann’s construction [2; Ch. 8, pg. 219].
- Coxeter-Todd \(K_{12}\) lattice— Applying Construction \(B_c\) to the ternary repetition code of length \(6\) over the Eisenstein integers yields the Coxeter-Todd \(K_{12}\) lattice [2; Ch. 7, pg. 200].
- \(E_6\) root lattice— The \([3,1,3]_3\) ternary repetition code can be used to obtain the \(E_6\) root lattice [3; Exam. 10.5.4][2; Ch. 7, pg. 200].
Member of code lists
- Algebraic-geometry codes
- Classical codes
- Classical codes with a rate
- Classical codes with notable decoders
- Cyclic codes
- Evaluation codes
- Locally correctable codes and friends
- Locally recoverable codes
- MDS codes and generalizations
- Non-binary linear codes
- Orthogonal arrays and friends
- Reed-Solomon codes and friends
- Universally optimal codes
Primary Hierarchy
Parents
\(q\)-ary repetition codes can be thought of as extended RS codes [4]. GRM\(_q(0,m)\) codes are evaluations of all zero-degree polynomials on \(\mathbb{F}_q^m\), which are just the \(q\) constant polynomials, so \(q\)-ary repetition codes are GRM\(_q(0,m)\) codes.
Reed-Solomon (RS) codeGRS MDS LRC Distributed-storage Evaluation AG Linear \(q\)-ary OA Universally optimal ECC \(t\)-design
\(q\)-ary repetition codes can be thought of as RS codes [4].
The \(q\)-ary repetition code is cyclic with generator polynomial \(1+x+\cdots+x^{n-1}\).
The \(q\)-ary repetition code is a \(q\)-ary sharp configuration [5; Table 12.1].
\(q\)-ary repetition code
Children
\(q\)-ary repetition code reduce to repetition codes for \(q=2\).
References
- [1]
- L. Golowich and V. Guruswami, “Quantum Locally Recoverable Codes”, (2023) arXiv:2311.08653
- [2]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [3]
- T. Ericson and V. A. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
- [4]
- 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
- [5]
- 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 (2022-12-12) — most recent
Cite as:
“\(q\)-ary repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_repetition