Reed-Muller Codes

Abstract

Reed-Muller codes were introduced in 1954, with a simple explicit construction based on polynomial evaluations, and have long been conjectured to achieve Shannon capacity on symmetric channels. Major progress was made towards a proof over the last decades; using combinatorial weight enumerator bounds, a breakthrough on the erasure channel from sharp thresholds, hypercontractivity arguments, and polarization theory. Another major progress recently established that the bit error probability vanishes slowly below capacity. However, when channels allow for errors, the results of Bourgain-Kalai do not apply for converting a vanishing bit to a vanishing block error probability, neither do the known weight enumerator bounds. The conjecture that RM codes achieve Shannon capacity on symmetric channels, with high probability of recovering the codewords, has thus remained open.This paper closes the conjecture's proof. It uses a new recursive boosting framework, which aggregates the decoding of codeword restrictions on 'subspace-sunflowers', handling their dependencies via an L p Boolean Fourier analysis, and using a list-decoding argument with a weight enumerator bound from Sberlo-Shpilka. The proof does not require a vanishing bit error probability for the base case, but only a non-trivial probability, obtained here for general symmetric codes. This gives in particular a shortened and tightened argument for the vanishing bit error probability result of Reeves-Pfister, and with prior works, it implies the strong wire-tap secrecy of RM codes on pure-state classical-quantum channels.

Reed–Muller Codes on BMS Channels Achieve Vanishing Bit-Error Probability for all Rates Below Capacity

“…If the input distribution has a doubly transitive symmetry, then this gives a bound that holds uniformly for all pairs of indices. Combining this result with the bounds in (34c) and (35) gives the single term bound stated in (28).…”

“…Note: After this paper was accepted for publication, an arXiv preprint [28] was posted with an argument that RM codes achieve vanishing block error probability for all rates below capacity. This new work builds on our observation (see Lemmas 8 and 9) that long RM codes can be punctured, in many different ways, down to much shorter RM codes of roughly the same rate.…”

“…Then, as a second stage, the decoder finds the nearest codeword to the first output sequence. The key technical achievement in [28] is showing that the BER of the first output sequence decays fast enough so that the second output equals the transmitted codeword with high probability.…”

Smoothing of Binary Codes, Uniform Distributions, and Applications

“…Proof. Very recently, Abbe and Sandon [53], building upon the work of Reeves and Pfister [54], proved that RM codes achieve capacity in symmetric channels. Therefore, the condition R b < 1 − h(δ b ) guarantees reliability.…”