Abstract
This paper considers ' \delta -Almost Reed-Muller codes', i.e., linear codes spanned by evaluations of all but a \delta fraction of monomials of degree at most d. It is shown that for any \delta > 0 and any \varepsilon >0 , there exists a family of \delta -Almost Reed-Muller codes of constant rate that correct 1/2-\varepsilon fraction of random errors with high probability. For exact Reed-Muller codes, the analogous result is not known and represents a weaker version of the longstanding conjecture that Reed-Muller codes achieve capacity for random errors (Abbe-Shpilka-Wigderson STOC '15). Our proof is based on the recent polarization result for Reed-Muller codes, combined with a combinatorial approach to establishing inequalities between the Reed-Muller code entropies.
Original language | American English |
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Pages (from-to) | 8034-8050 |
Number of pages | 17 |
Journal | IEEE Transactions on Information Theory |
Volume | 67 |
Issue number | 12 |
DOIs | |
State | Published - 1 Dec 2021 |
Externally published | Yes |
Keywords
- polarization
- Reed-Muller codes
All Science Journal Classification (ASJC) codes
- Information Systems
- Computer Science Applications
- Library and Information Sciences