Description
Extension of the \([7,4,3]\) Hamming code by a parity-check bit. The smallest doubly even self-dual code.Cousins
- 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 [4; Exam. 10.5.2].
- \([7,4,3]\) 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.
- Klemm code— The binary image of the \(m=1\) Klemm code under the Gray map is the \([8,4,4]\) extended Hamming code [5; Exam. 3.2].
- Octacode— The mod-two reduction of the octacode is the \([8,4,4]\) extended Hamming code [6].
- \([[8, 3, 3]]\) 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
\([2^r,2^r-r-1,4]\) Extended Hamming codeQR Linear \(q\)-ary Linear code over \(\mathbb{Z}_q\) Cyclic Constacyclic Skew-cyclic Small-distance block ECC
Parents
\([2^m,m+1,2^{m-1}]\) 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 the smallest doubly even self-dual code.
Binary quadratic-residue (QR) codeQR Linear \(q\)-ary Linear code over \(\mathbb{Z}_q\) Cyclic Constacyclic Skew-cyclic ECC
The \([8,4,4]\) extended Hamming code is an extended quadratic-residue code [7].
\([8,4,4]\) 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]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [5]
- Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
- [6]
- Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
- [7]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
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