Hexacode

Hexacode[1,2]

Alternative names: Golay code over \(GF(4)\).

Description

The \([6,3,4]_4\) self-dual MDS code that has connections to projective geometry, lattices [2], and conformal field theory [3]. Puncturing the code yields the perfect \([5,3,3]_4\) quaternary Hamming code known as the shortened hexacode or shorter hexacode [4]. Both codes are sometimes refereed to as Golay codes over \(GF(4)\).

A generator matrix for the hexacode is \begin{align} \begin{pmatrix} 1 & 0 & 0 & 1 & 1 & \omega\\ 0 & 1 & 0 & 1 & \omega & 1\\ 0 & 0 & 1 & \omega & 1 & 1 \end{pmatrix}~, \tag*{(1)}\end{align} where \(GF(4) = \{0,1,\omega, \bar{\omega}\}\) is the quaternary Galois field.

Decoding

Bounded-distance decoder requiring at most 34 real operations [5].

Notes

See [6; Sec. 10.3] for an exposition.See corresponding MinT database entry [7].

Cousins

\(q\)-ary Hamming code— The hexacode is an extended quaternary Hamming code [6; Exer. 578].
Five-qubit perfect code— Applying the qubit Hermitian construction to the hexacode yields a \([[6,0,4]]\) quantum code [8] corresponding to the six-qubit AME state. The five-qubit code can be obtained either by applying the qubit Hermitian construction to the shortened hexacode [9; Exam. A] or by tracing out a qubit of the \([[6,0,4]]\) code [10; Appx. A].
\([24, 12, 8]\) Extended Golay code— Extended Golay codewords can be obtained from hexacodewords [2]. The hexacode can be used to decode the extended Golay code [11].
\([23, 12, 7]\) Golay code— There a connection between automorphisms of the even Golay code and the holomorph of the hexacode [3]. The hexacoed is often referred to as the Golay code over \(GF(4)\) [6].
Perfect code— The shortened hexacode is perfect [6; Exer. 578].
Reed-Solomon (RS) code— The dual of the shortened hexacode code is a \([5,2,4]_4\) doubly extended RS code [9; Exam. A].
Spherical design— The hexacode is a complex spherical 3-design when embedded into the complex sphere via the polyphase mapping [12].
Polyphase code— The hexacode is a complex spherical 3-design when embedded into the complex sphere via the polyphase mapping [12].
Coxeter-Todd \(K_{12}\) lattice— The hexacode can be used to obtain the Coxeter-Todd \(K_{12}\) lattice [13; Exam. 10.5.6].

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

\(q\)-ary codeECC

Additive \(q\)-ary codeECC

Linear \(q\)-ary codeECC

Projective geometry code

Hyperoval code

Parents

Hyperoval code

Columns of hexacode's generator matrix represent the six points of a hyperoval in the projective plane \(PG(2,4)\) [14; pg. 289][15; Exam. 19.2.1].

Evaluation AG codeEvaluation Linear \(q\)-ary AG ECC

The hexacode is an evaluation AG code over the quaternary Galois field \(GF(4) = \{0,1,\omega, \bar{\omega}\}\) with \(\cal X\) defined by \(x^2 y + \omega y^2 z + \bar{\omega} z^2 x = 0\) [16; Exam. 2.77].

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

The hexacode is the smallest example of an extended quadratic-residue code of Type \(4^H\) [17; Sec. 2.4.6][6; Exer. 363]. The shortened hexacode is an odd-like quadratic-residue code [6; Exam. 6.6.8].

Self-dual linear codeSelf-dual additive Linear \(q\)-ary ECC

The hexacode is Hermitian self-dual and, as a result, is also trace-Hermitian self-dual additive [6; Sec. 9.10]. The hexacode and the shortened hexacode are extremal [6; Tab. 9.14][4; Tm. 12].

Denniston codeMDS Projective geometry Linear \(q\)-ary OA LRC Distributed-storage Sharp configuration \(t\)-design Universally optimal ECC

A version of the hexacode is recovered for Dennison code parameters \(i=1\) and \(a=2\) [14].

Maximum distance separable (MDS) codeLinear \(q\)-ary OA LRC Distributed-storage \(t\)-design Universally optimal ECC

The hexacode is an MDS code [6; Exer. 578].

Extended GRS codeGRM GRS Evaluation MDS Linear \(q\)-ary OA AG \(t\)-design Universally optimal LDC LCC LRC Distributed-storage ECC

The hexacode is an extended RS code [2; pg. 82].

Lexicographic codeECC

Hexacodewords can be arranged in an order from smallest to largest, with each codeword differing at four places from the next [18][19; pg. 327].

Small-distance block codeECC

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]: J. A. Harvey and G. W. Moore, “Moonshine, superconformal symmetry, and quantum error correction”, Journal of High Energy Physics 2020, (2020) arXiv:2003.13700 DOI
[4]: G. Hoehn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266
[5]: A. Vardy, “Even more efficient bounded-distance decoding of the hexacode, the Golay code, and the Leech lattice”, IEEE Transactions on Information Theory 41, 1495 (1995) DOI
[6]: W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[7]: Rudolf Schürer and Wolfgang Ch. Schmid. “Hexacode.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CHexa.html
[8]: A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
[9]: 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
[10]: 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
[11]: V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986) DOI
[12]: V. V. Albert, private communication, 2024.
[13]: T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[14]: J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
[15]: A. E. Brouwer, "Two-weight Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[16]: T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
[17]: Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
[18]: R. A. Wilson, On lexicographic codes of minimal distance 4, Atti Sem. Mat. Fis. Univ. Modena 33 (1984)
[19]: F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.

Page edit log

Victor V. Albert (2022-08-09) — most recent

Cite as:

“Hexacode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hexacode

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