\([7,4,3]\) Hamming code[1][2][3]

Description

Second-smallest member of the Hamming code family with generator matrix \begin{align} \left(\begin{array}{ccccccccccc} 1 & 0 & 0 & 0 & 1 & 1 & 0\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \end{array}\right)~. \tag*{(1)}\end{align}

This 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. It can be shortened to yield the \([6,3,3]\) shortened Hamming code.

Protection

Can detect 1-bit and 2-bit errors, and can correct 1-bit errors.

Parents

  • Hamming code
  • Projective geometry code — The \([7,4,3]\) Hamming code parity-check matrix corresponds to points in the Fano plane \(PG_2(2)\). Columns of a general Hamming parity-check matrix correspond to one-dimensional subspaces of \(GF(2)^n\).
  • Binary quadratic-residue (QR) code — \([7,4,3]\) Hamming code is a quadratic-residue code with generator polynomial \(1+x+x^3\) [4].

Cousins

  • \([[7,1,3]]\) Steane code — The Steane code is constructed from the \([7,4,3]\) classical Hamming code.
  • Divisible code — The extended Hamming code code is the smallest double-even self-dual code.
  • Griesmer code — Starting with the \([6,3,3]\) shortened Hamming code and applying the \((u|u+v)\) construction recursively using the repetition code yields a family of \([2^m,m+1,2^{m-1}]\) codes for \(m\geq1\) that saturate the Griesmer bound [5; pg. 90].
  • \(E_8\) Gosset lattice code — The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice code via the mod-two lattice construction [6; Ex. 10.5.2].
  • \((u|u+v)\)-construction code — Starting with the \([6,3,3]\) shortened Hamming code and applying the \((u|u+v)\) construction recursively using the repetition code yields a family of \([2^m,m+1,2^{m-1}]\) codes for \(m\geq1\) that saturate the Griesmer bound [5; pg. 90].
  • Octacode — The octacode reduces modulo-two to the \([8,4,4]\) extended Hamming code [7].

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]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[5]
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
[6]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[7]
Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
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Zoo Code ID: hamming743

Cite as:
\([7,4,3]\) Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamming743
BibTeX:
@incollection{eczoo_hamming743, title={\([7,4,3]\) Hamming code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hamming743} }
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Cite as:

\([7,4,3]\) Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamming743

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/bits/easy/hamming743.yml.