Alternative Names: \([8,4,4]\) \(e_8\) code.
Description
Extension of the \([7,4,3]\) Hamming code by a parity-check bit. The smallest doubly even self-dual code, and the unique Type II code of length \(8\) [4; Rem. 4.3.10].
A generator matrix is \begin{align} \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \end{pmatrix}~, \tag*{(1)}\end{align} equivalent to the standard form \begin{align} \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 \end{pmatrix}~. \tag*{(2)}\end{align} Its automorphism group is \(GA(3,\mathbb{F}_2)\) [5].
Cousins
- Group-algebra code— The \([8,4,4]\) extended Hamming code is a group-algebra code for the group \(\mathbb{Z}_2 \times \mathbb{Z}_4\) [5].
- Binary quadratic-residue (QR) code— The \([8,4,4]\) extended Hamming code is an extended quadratic-residue code [6].
- Divisible code— The \([8,4,4]\) extended Hamming code is the smallest doubly even self-dual code, and the unique Type II code of length \(8\) [4; Rem. 4.3.10].
- \(E_8\) Gosset lattice— The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice via Construction A [7; Exam. 10.5.2][8; pg. 138].
- \([24, 12, 8]\) Extended Golay code— The extended Golay code can be constructed from two suitably chosen extended Hamming \([8,4,4]\) codes using the \(|a+x|b+x|a+b+x|\) construction [6; pg. 588].
- \([7,4,3]\) Hamming code— The Hamming code can be extended by a parity-check bit to yield the \([8,4,4]\) extended Hamming code, the smallest doubly even self-dual code.
- \([4,2,3]_4\) RS\(_4\) code— The RS\(_4\) code can be mapped to the \([8,4,4]\) extended Hamming code [9; Sec. 2.4.2] by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) [10].
- Klemm code— The binary image of the \(m=1\) Klemm code under the Gray map is the \([8,4,4]\) extended Hamming code [11; Exam. 3.2].
- Octacode— The mod-two reduction of the octacode is the \([8,4,4]\) extended Hamming code [9]. The octacode can be obtained by Hensel-lifting the \([8,4,4]\) extended Hamming code to \(\mathbb{Z}_4\) [12].
- \([[8,3,2]]\) Smallest interesting color code— The \([[8,3,2]]\) hypercube code \(H_X\) check matrix is the parity-check matrix of the \([8,4,4]\) extended Hamming code, while its \(H_Z\) matrix is that of the SPC code.
- \([[8, 3, 3]]\) Eight-qubit Gottesman code— The \([[8, 3, 3]]\) code is obtained via a modified CSS construction from the \([8,4,4]\) extended Hamming code.
Primary Hierarchy
Parents
The \([8,4,4]\) extended Hamming code is the smallest doubly even self-dual code, and the unique Type II code of length \(8\) [4; Rem. 4.3.10].
\([2^m,2^m-m-1,4]\) Extended Hamming codeLinear code over \(\mathbb{Z}_q\) GRM Evaluation Divisible Linear \(q\)-ary AG LCC LRC Distributed-storage Small-distance block ECC
\([2^m,m+1,2^{m-1}]\) First-order RM codeLinear code over \(\mathbb{Z}_q\) GRM Evaluation Divisible Linear \(q\)-ary AG LCC LRC Distributed-storage ECC
The \([8,4,4]\) extended Hamming code is the first-order RM\((1,3)\) code.
\([8,4,4]\) extended Hamming code
References
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- M. J. E. Golay, “Notes on digital coding”, Proceedings of the IEEE 37, 657 (1949)
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- S. Bouyuklieva, “Self-dual codes”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 79-96 DOI
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- M. Borello and W. Willems, “On the algebraic structure of quasi group codes”, (2021) arXiv:1912.09167
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- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
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- T. Ericson and V. A. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
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- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
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- Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
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- P. Gaborit, V. Pless, P. Solé, and O. Atkin, “Type II Codes over F4”, Finite Fields and Their Applications 8, 171 (2002) DOI
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- Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
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- A. R. Calderbank and N. J. A. Sloane, “Modular and p-adic cyclic codes”, (2003) arXiv:math/0311319
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2022-12-21)
Cite as:
“\([8,4,4]\) extended Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/hamming844, arXiv:2606.11484