\([5,3,3]_4\) Shortened hexacode

[Jump to code hierarchy]

\([5,3,3]_4\) Shortened hexacode[1,2]

Alternative Names: Shorter hexacode, Golay code over \(\mathbb{F}_4\).

Description

A perfect \([5,3,3]_4\) quaternary Hamming code that is the result of puncturing the hexacode [3].

Cousins

\([6,3,4]_4\) Hexacode— The shortened hexacode is obtained by puncturing the hexacode.
Self-dual linear code— The hexacode and the shortened hexacode are extremal [4; Tab. 9.14][3; Tm. 12].
\([[5,1,3]]\) Five-qubit perfect code— The five-qubit code can be obtained either by applying the qubit Hermitian construction to the shortened hexacode [5; Exam. A] or by tracing out a qubit of the \([[6,0,4]]\) code [6; Appx. A].
\([23, 12, 7]\) Golay code— The shortened hexacode is often referred to as the Golay code over \(\mathbb{F}_4\) [4].
Reed-Solomon (RS) code— The dual of the shortened hexacode code is a \([5,2,4]_4\) doubly extended RS code [5; Exam. A].
\([[5,1,3]]_4\) Galois-qudit CSS code— The \([[5,1,3]]_4\) code is obtained from the shortened hexacode [7].

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

\(q\)-ary codeECC

Completely regular code

Perfect codeECC

Parents

Perfect code

The shortened hexacode is perfect [4; Exer. 578].

Quadratic-residue (QR) codeLinear \(q\)-ary Cyclic Constacyclic Skew-cyclic ECC

The shortened hexacode is an odd-like quadratic-residue code [4; Exam. 6.6.8].

Extended GRS codeGRM Evaluation Linear \(q\)-ary AG LCC LRC Distributed-storage ECC

The shortened hexacode is a doubly extended narrow-sense RS code [2; pg. 82].

Small-distance block codeECC

\([5,3,3]_4\) Shortened hexacode

References

[1]: K. A. Bush, “Orthogonal Arrays of Index Unity”, The Annals of Mathematical Statistics 23, 426 (1952) DOI
[2]: J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[3]: G. Höhn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266
[4]: W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[5]: G. D. Forney, M. Grassl, and S. Guha, “Convolutional and Tail-Biting Quantum Error-Correcting Codes”, IEEE Transactions on Information Theory 53, 865 (2007) arXiv:quant-ph/0511016 DOI
[6]: F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015) arXiv:1503.06237 DOI
[7]: 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

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2025-11-06)

Cite as:

“\([5,3,3]_4\) Shortened hexacode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/shortened_hexacode, arXiv:2606.11484

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