\([[6,2,3]]_{q}\) code

\([[6,2,3]]_{q}\) code[1,2]

Description

Six-qudit MDS error-correcting code defined for Galois-qudit dimension \(q=3\) [1], \(q=2^2\) [3], and \(q \geq 5\) [1][2; Exam. 33]. This code cannot exist for qubits (\(q=2\)).

Encoding

Three different encoding circuits for \(q=3\) [4].

Cousin

Graph quantum code— The \([[6,2,3]]_{q}\) code family contains examples of graph quantum codes [1].

Member of code lists

Primary Hierarchy

Quantum Domain

Galois-qudit Kingdom

Galois-qudit codeQECC Quantum

Galois-qudit USt code

Galois-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum

True Galois-qudit stabilizer code

Parents

True Galois-qudit stabilizer code

The code is a non-CSS stabilizer code in general [3].

Quantum maximum-distance-separable (MDS) codeQECC Quantum

Small-distance block quantum codeQECC Quantum

\([[6,2,3]]_{q}\) code

References

[1]: Keqin Feng, “Quantum codes [[6, 2, 3]]/sub p/ and [[7, 3, 3]]/sub p/ (p ≥ 3) exist”, IEEE Transactions on Information Theory 48, 2384 (2002) DOI
[2]: A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
[3]: Z. Wang, S. Yu, H. Fan, and C. H. Oh, “Quantum error-correcting codes over mixed alphabets”, Physical Review A 88, (2013) arXiv:1205.4253 DOI
[4]: M. Grassl, “Variations on Encoding Circuits for Stabilizer Quantum Codes”, Lecture Notes in Computer Science 142 (2011) DOI

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2024-03-01)

Cite as:

“\([[6,2,3]]_{q}\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/galois_6_2_3, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/small/galois_6_2_3.yml.