Reed–Muller code

Reed–Muller codes are error-correcting codes that are used in wireless communications applications, particularly in deep-space communication. Moreover, the proposed 5G standard relies on the closely related polar codes for error correction in the control channel. Due to their favorable theoretical and mathematical properties, Reed–Muller codes have also been extensively studied in theoretical computer science.

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 generalize the Reed–Solomon codes and the Walsh–Hadamard code. Reed–Muller codes are linear block codes that are locally testable, locally decodable, and list decodable. These properties make them particularly useful in the design of probabilistically checkable proofs.

Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. When r and m are integers with 0 ≤ r ≤ m, the Reed–Muller code with parameters r and m is denoted as RM(r, m). When asked to encode a message consisting of k bits, where ${\displaystyle \textstyle k=\sum _{i=0}^{r}{\binom {m}{i}}}$ holds, the RM(r, m) code produces a codeword consisting of 2^m bits.

Reed–Muller codes are named after David E. Muller, who discovered the codes in 1954, and Irving S. Reed, who proposed the first efficient decoding algorithm.