Reed-Muller codes for random erasures and errors
Emmanuel Abbe, Amir Shpilka, Avi Wigderson

TL;DR
This paper analyzes the capacity of Reed-Muller codes to correct random erasures and errors, providing new bounds and decoding techniques that demonstrate their optimal performance in various regimes.
Contribution
It introduces novel bounds and techniques showing Reed-Muller codes achieve capacity for erasures and errors across different rate regimes, including a new reduction for error decoding.
Findings
RM codes achieve capacity for erasures at high and low rates
RM codes can uniquely decode at about square root of errors at high rates
New bounds on weight distribution of RM codes
Abstract
This paper studies the parameters for which Reed-Muller (RM) codes over can correct random erasures and random errors with high probability, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate polynomials on random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity. The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about , the matrix whose rows are truth tables of all monomials…
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Taxonomy
TopicsCoding theory and cryptography · DNA and Biological Computing · Error Correcting Code Techniques
