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.
A generator matrix is \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*{(1)}\end{align}
Its automorphism group is \(GA(3,\mathbb{F}_2)\) [4].
Cousins
- Binary quadratic-residue (QR) code— The \([8,4,4]\) extended Hamming code is an extended quadratic-residue code [5].
- Divisible code— The \([8,4,4]\) extended Hamming code code is the smallest double-even self-dual code.
- \(E_8\) Gosset lattice— The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice via Construction A [6; Exam. 10.5.2][7; pg. 138].
- \([7,4,3]\) 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.
- Reed-Solomon (RS) code— The \([4,2,3]_4\) RS code is the smallest example of a quaternary quadratic-residue code and can be mapped to the \([8,4,4]\) extended Hamming code [8; Sec. 2.4.2] by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) [9].
- Klemm code— The binary image of the \(m=1\) Klemm code under the Gray map is the \([8,4,4]\) extended Hamming code [10; Exam. 3.2].
- Octacode— The mod-two reduction of the octacode is the \([8,4,4]\) extended Hamming code [8].
- \([[8, 3, 3]]\) 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.
\([2^m,2^m-m-1,4]\) Extended Hamming codeSelf-dual linear Self-dual additive GRM Evaluation Divisible Linear \(q\)-ary Linear code over \(\mathbb{Z}_q\) LCC LRC Distributed-storage LDC Small-distance block ECC
\([2^m,m+1,2^{m-1}]\) First-order RM codeGRM Evaluation Divisible Linear \(q\)-ary Linear code over \(\mathbb{Z}_q\) LCC LRC Distributed-storage LDC ECC
The \([8,4,4]\) extended Hamming code is a first-order RM code because it is self-dual and first-order RM codes are dual to extended Hamming codes.
The \([8,4,4]\) extended Hamming code is a group-algebra code for the group \(\mathbb{Z}_2 \times \mathbb{Z}_4\) [4].
\([8,4,4]\) extended Hamming code
References
- [1]
- C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948) DOI
- [2]
- R. W. Hamming, “Error Detecting and Error Correcting Codes”, Bell System Technical Journal 29, 147 (1950) DOI
- [3]
- M. J. E. Golay, Notes on digital coding, Proc. IEEE, 37 (1949) 657.
- [4]
- M. Borello and W. Willems, “On the algebraic structure of quasi group codes”, (2021) arXiv:1912.09167
- [5]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
- [6]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [7]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [8]
- Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
- [9]
- P. Gaborit, V. Pless, P. Solé, and O. Atkin, “Type II Codes over F4”, Finite Fields and Their Applications 8, 171 (2002) DOI
- [10]
- Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
Page edit log
- Victor V. Albert (2022-12-21) — most recent
Cite as:
“\([8,4,4]\) extended Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamming844