## Abstract

This paper studies the parameters for which binary Reed-Muller (RM) codes can be decoded successfully on the BEC and BSC, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate GF(2) 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 E(m,r), the matrix whose rows are truth tables of all monomials of degree ≤ r in m variables. What is the most (resp. least) number of random columns in E(m, r) that define a subma-trix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees r, which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code C of sufficiently high rate we construct a new code C' obtained by tensoring C, such that for every subset S of coordinates, if C can recover from erasures in S, then C' can recover from errors in S. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent [27] bounds from constant degree to linear degree polynomials.

Original language | English |
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Title of host publication | STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing |

Publisher | Association for Computing Machinery |

Pages | 297-306 |

Number of pages | 10 |

ISBN (Electronic) | 9781450335362 |

DOIs | |

State | Published - 14 Jun 2015 |

Event | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 - Portland, United States Duration: 14 Jun 2015 → 17 Jun 2015 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | 14-17-June-2015 |

ISSN (Print) | 0737-8017 |

### Conference

Conference | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 |
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Country/Territory | United States |

City | Portland |

Period | 14/06/15 → 17/06/15 |

## Keywords

- Channel capacity
- Multivariate polynomials
- Reed-Muller codes
- Weight enumerator