Reed–Muller Code
The binary Reed-Muller code RM(r, u) with 0 ≤ r ≤ u, due to [1] and [2], is a linear [2u, n, 2u−r]-code over ℤ2 with
+ 
+ ⋯ + 
. 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(2n, 2u, ℤ2, 2r+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 [2u, 1, 2u]-repetition code.
RM(u−1, u) is the [2u, 2u − 1, 2]-parity-check code.
RM(u, u) is the [2u, 2u, 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: 2024-09-05.
http://mint.sbg.ac.at/desc_CReedMuller.html