Reed-Muller codes are error-correcting codes that are used in wireless communications applications, particularly in deep-space communication.
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 codes are named after David E. Muller, who discovered the codes in 1954, and Irving S. Reed, who proposed the first efficient decoding algorithm.
Traditional Reed-Muller codes are binary codes, which means that messages and codewords are binary strings. When $r$ and $m$ are integers with $0 \leq r \leq m,$ 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 $k=\sum _{i=0}^{r}{\binom {m}{i}}$ holds, the $RM(r, m)$ code produces a codeword consisting of $2^m$ bits.