61 research outputs found
Enhanced Recursive Reed-Muller Erasure Decoding
Recent work have shown that Reed-Muller (RM) codes achieve the erasure
channel capacity. However, this performance is obtained with maximum-likelihood
decoding which can be costly for practical applications. In this paper, we
propose an encoding/decoding scheme for Reed-Muller codes on the packet erasure
channel based on Plotkin construction. We present several improvements over the
generic decoding. They allow, for a light cost, to compete with
maximum-likelihood decoding performance, especially on high-rate codes, while
significantly outperforming it in terms of speed
Reed-Muller codes have vanishing bit-error probability below capacity: a simple tighter proof via camellia boosting
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 into `camellias', 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
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