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\([4,2,3]_4\) RS\(_4\) code

Alternative names: \(XQ(\mathbb{F}_4,3)\).

Description

A Type II Euclidean self-dual extended RS code that is the smallest quaternary extended QR code [1; pg. 296][2; Sec. 2.4.2]. Puncturing the \([4,2,3]_4\) RS\(_4\) code yields the \([3,2,2]_4\) shortened RS\(_4\) code, which is an RS code [1; pg. 295].

A generator matrix for the code is \begin{align} \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & \omega & \omega^2 \end{pmatrix}~, \tag*{(1)}\end{align} where \(\mathbb{F}_4 = \{0,1,\omega, \bar{\omega}\}\) is the quaternary Galois field.

The automorphism group of the code is \(3.S_4\) [2; Sec. 2.4.2].

Cousins

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Dual linear codeECC
Self-dual linear codeSelf-dual additive ECC
Parents
Self-dual linear code
The RS\(_4\) is the smallest Type II Euclidean self-dual code [2; Sec. 2.4.2].
Extended GRS codeGRM Evaluation Linear \(q\)-ary AG LCC LRC Distributed-storage ECC
The RS\(_4\) is an extended RS code [1; pg. 296].
Maximum distance separable (MDS) codeLRC Distributed-storage Linear \(q\)-ary OA Universally optimal ECC \(t\)-design
Small-distance block codeECC
\([4,2,3]_4\) RS\(_4\) code

References

[1]
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
[2]
Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
[3]
P. Gaborit, V. Pless, P. Solé, and O. Atkin, “Type II Codes over F4”, Finite Fields and Their Applications 8, 171 (2002) DOI
[4]
J. M. Koh, A. Gong, A. C. Diaconu, D. B. Tan, A. A. Geim, M. J. Gullans, N. Y. Yao, M. D. Lukin, and S. Majidy, “Entangling logical qubits without physical operations”, (2026) arXiv:2601.20927
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Zoo Code ID: reed_solomon_4

Cite as:
“\([4,2,3]_4\) RS\(_4\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/reed_solomon_4
BibTeX:
@incollection{eczoo_reed_solomon_4, title={\([4,2,3]_4\) RS\(_4\) code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/reed_solomon_4} }
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Cite as:

“\([4,2,3]_4\) RS\(_4\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/reed_solomon_4

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/easy/reed_solomon_4.yml.