Reed–Muller code

Reed-Muller code RM(r,m)
Named after	Irving S. Reed and David E. Muller
Classification
Type	Linear block code
Block length	${\displaystyle 2^{m}}$
Message length	${\displaystyle k=\sum _{i=0}^{r}{\binom {m}{i}}}$
Rate	${\displaystyle k/2^{m}}$
Distance	${\displaystyle 2^{m-r}}$
Alphabet size	${\displaystyle 2}$
Notation	${\displaystyle [2^{m},k,2^{m-r}]_{2}}$-code

Reed–Muller codes are a family of linear error-correcting codes used in communications. Reed–Muller codes belong to the classes of locally testable codes and locally decodable codes, which is why they are useful in the design of probabilistically checkable proofs in computational complexity theory. They are named after Irving S. Reed and David E. Muller. Muller discovered the codes, and Reed proposed the majority logic decoding for the first time. Special cases of Reed–Muller codes include the Walsh–Hadamard code and the Reed–Solomon code.

Reed–Muller codes are listed as RM(r, m), where r is the order of the code, 0 ≤ r ≤ m, and m determines the block length N = 2^m. RM codes are related to binary functions on the field GF(2^m) over the elements {0, 1}.

RM(0, m) codes are repetition codes of length N = 2^m, rate ${\displaystyle {R={\tfrac {1}{N}}}}$ and minimum distance ${\displaystyle d_{\min }=N}$.

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