\([7,4,3]\) Hamming code[13] 

Description

Second-smallest member of the Hamming code family.

Its generator matrix is \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} Up to equivalence, this is the only nontrivial length-seven perfect binary code containing the zero vector.

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. It can be shortened to yield the \([6,3,3]\) shortened Hamming code. The dual of the Hamming code is its even-weight subcode, the \([7,3,4]\) little Hamming code, also known as the \(S(2,3)\) simplex code [4,5].

Protection

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

Parents

Cousins

  • Incidence-matrix projective code — The \([7,4,3]\) Hamming code parity-check matrix corresponds to points in the Fano plane \(PG_2(2)\) [7; Ex. 21.4.2].
  • \([8,4,4]\) extended 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.
  • Dual linear code — The dual of the Hamming code is its even-weight subcode, the \([7,3,4]\) little Hamming code, also known as the \(S(2,3)\) simplex code [4,5].
  • \([[7,1,3]]\) Steane code — The Steane code is constructed from the \([7,4,3]\) classical Hamming code via the CSS construction.
  • 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 [8; pg. 90].
  • \(E_7\) root lattice code — The \([7,4,3]\) Hamming code yields the \(E_7^{\perp}\) root lattice code via Construction A [9]. The \([7,3,4]\) little Hamming code yields the \(E_7\) root lattice code via the same construction [9][10; Ex. 10.5.3].
  • Combinatorial design — Weight-three and weight-four codewords of the \([7,4,3]\) Hamming code support combinatorial \(2\)-\((7,3,1)\) and \(2\)-\((7,4,2)\) designs, respectively [11; Ex. 5.2.5].
  • \([7,3,4]\) simplex code — The \([7,4,3]\) Hamming and \([7,3,4]\) simplex codes are dual to each other.
  • \((u|u+v)\)-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 [8; pg. 90].
  • Octacode — The octacode can be obtained by Hensel-lifting the \([7,4,3]\) Hamming code to \(\mathbb{Z}_4\) [12].

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]
W. Feit. Some remarks on weight functions of spaces over GF(2), unpublished (1972)
[5]
C. L. Mallows and N. J. A. Sloane, “Weight enumerators of self-orthogonal codes”, Discrete Mathematics 9, 391 (1974) DOI
[6]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[7]
C. A. Kelley, "Codes over Graphs." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[8]
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
[9]
J. H. Conway and N. J. A. Sloane, “On the Voronoi Regions of Certain Lattices”, SIAM Journal on Algebraic Discrete Methods 5, 294 (1984) DOI
[10]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[11]
V. D. Tonchev, "Codes and designs." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[12]
A. R. Hammons et al., “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) 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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\([7,4,3]\) 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/edit/main/codes/classical/bits/easy/hamming/hamming743.yml.