Reed–Muller Code

The binary Reed-Muller code RM(r, u) with 0 ≤ ru, due to [1] and [2], is a linear [2u, n, 2ur]-code over 2 with

n = $\displaystyle \binom{u}{0}$ + $\displaystyle \binom{u}{1}$ + ⋯ + $\displaystyle \binom{u}{r}$.

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(ur−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

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

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