## Reed–Muller Code

The binary Reed-Muller code RM(*r*, *u*) with 0 ≤ *r* ≤ *u*, due to [1] and [2], is a linear [2^{u}, *n*, 2^{u−r}]-code over ℤ_{2} with

*n*= + + ⋯ + .

The parameter *r* is called the order of RM(*r*, *u*). For *r* < *u* the dual code of RM(*r*, *u*) is again a Reed-Muller Code, namely RM(*u*−*r*−1, *u*). Therefore RM(*r*, *u*) is also an OA(2^{n}, 2^{u}, ℤ_{2}, 2^{r+1} − 1).

For 0 < *r* < *u* RM(*r*, *u*) can be obtained recursively by applying the (*u*, *u* + *v*)-construction to RM(*r*−1, *u*−1) and RM(*r*, *u*−1).

### Special Cases

RM(0,

*u*) is the [2^{u}, 1, 2^{u}]-repetition code.RM(

*u*−1,*u*) is the [2^{u}, 2^{u}− 1, 2]-parity-check code.RM(

*u*,*u*) is the [2^{u}, 2^{u}, 1]-code without redundancy.

### See also

### References

[1] | D. E. Muller. Application of Boolean algebra to switching circuit design and to error detection. IRE Transactions on Computers, 3:6–12, 1954. |

[2] | Irving S. Reed. A class of multiple-error-correcting codes and the decoding scheme. IEEE Transactions on Information Theory, 4:38–49, 1954. |

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

[4] | 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. “Reed–Muller Code.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CReedMuller.html