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 automorphism group of the code is \(GL_{3}(\mathbb{F}_{2})\), the second-smallest simple group.
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
Parents
- \([2^r-1,2^r-r-1,3]\) Hamming code
- Binary quadratic-residue (QR) code — \([7,4,3]\) Hamming code is a quadratic-residue code with generator polynomial \(1+x+x^3\) [6].
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 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].
- \((u|u+v)\)-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
Page edit log
- Victor V. Albert (2022-12-21) — most recent
Cite as:
“\([7,4,3]\) Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamming743