Reed-Muller codes have vanishing bit-error probability below capacity: a simple tighter proof via camellia boosting
Abstract: This paper shows that a class of codes such as Reed-Muller (RM) codes have vanishing bit-error probability below capacity on symmetric channels. The proof relies on the notion of camellia codes': a class of symmetric codes decomposable intocamellias', i.e., set systems that differ from sunflowers by allowing for scattered petal overlaps. The proof then follows from a boosting argument on the camellia petals with second moment Fourier analysis. For erasure channels, this gives a self-contained proof of the bit-error result in Kudekar et al.'17, without relying on sharp thresholds for monotone properties Friedgut-Kalai'96. For error channels, this gives a shortened proof of Reeves-Pfister'23 with an exponentially tighter bound, and a proof variant of the bit-error result in Abbe-Sandon'23. The control of the full (block) error probability still requires Abbe-Sandon'23 for RM codes.
- A proof that Reed-Muller codes achieve Shannon capacity on symmetric channels. In proc. FOCS, 2023.
- E. Friedgut and G. Kalai. Every monotone graph property has a sharp threshold. Proceedings of the American mathematical Society, 124(10):2993–3002, 1996.
- Reed–Muller codes achieve capacity on erasure channels. IEEE Transactions on Information Theory, 63(7):4298–4316, 2017.
- Reed–Muller codes on BMS channels achieve vanishing bit-error probability for all rates below capacity. IEEE Transactions on Information Theory, 2023.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.