## Hamming Code

Hamming codes H(*m*, *b*) (introduced in [1] and [2]) are linear single-error-correcting [*s*, *s*−*m*, 3]-codes over **F**_{b} with

*s*=

for all *m* ≥ 2.

The dual orthogonal array A = H(*m*, *b*)^{⊥} is a linear OA(*b*^{m}, *s*, *b*, 2) with strength 2, which was already known by Rao [3]. An *m*×*s* generator matrix of A (and parity check matrix of H(*m*, *b*)) is constructed by using the *s* pairwise linearly independent vectors of **F**_{b}^{m}. Such a set can be found, e.g. by choosing all non-zero vectors with leftmost non-zero coordinate equal to 1 from **F**_{b}^{m}. When interpreted as a code, A is the [*s*, *m*, *b*^{m−1}]-simplex code.

### Optimality

Hamming codes are perfect codes, since they meet the Hamming or sphere packing bound. Alternatively, their dual OAs are tight.

### See also

Hamming code at

Hamming code at

### References

[1] | Richard Wesley Hamming. Error detecting and error correcting codes. Bell System Technical Journal, 29:147–160, 1950. |

[2] | Marcel J. E. Golay. Notes on digital coding. Proceedings of the IEEE, 37:657, 1949. |

[3] | Calyampudi Radhakrishna Rao. On a class of arrangements. Proceedings of the Edinburgh Mathematical Society, 8:119–125, 1949. |

[4] | F. Jessie MacWilliams and Neil J. A. Sloane.The Theory of Error-Correcting Codes.North-Holland, Amsterdam, 1977. |

[5] | Jürgen Bierbrauer.Introduction to Coding Theory.Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004. MR2079734 (2005f:94001) |

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Hamming Code.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_CHamming.html