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

\([7,4,3]\) Hamming code[1–3]

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} 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. Its dual is the \([7,3,4]\) little Hamming code, also known as the \(S(2,3)\) simplex code.

Protection

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

Parents

Hamming code
Binary quadratic-residue (QR) code — \([7,4,3]\) Hamming code is a quadratic-residue code with generator polynomial \(1+x+x^3\) [4].

Cousins

Incidence-matrix projective code — The \([7,4,3]\) Hamming code parity-check matrix corresponds to points in the Fano plane \(PG_2(2)\) [5; 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.
\([[7,1,3]]\) Steane code — The Steane code is constructed from the \([7,4,3]\) classical Hamming 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 [6; pg. 90].
\(E_7\) root lattice code — The \([7,4,3]\) Hamming code yields the \(E_7^{\perp}\) root lattice code via the mod-two lattice construction [7]. The \([7,3,4]\) little Hamming code yields the \(E_7\) root lattice code via the same construction [7][8; Ex. 10.5.3].
Combinatorial design code — 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 [9; Ex. 5.2.5].
\((u|u+v)\)-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 [6; pg. 90].

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]: C. A. Kelley, "Codes over Graphs." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[6]: J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
[7]: 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
[8]: T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[9]: V. D. Tonchev, "Codes and designs." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) 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

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