1. Almost-Reed--Muller Codes Achieve Constant Rates for Random Errors
- Author
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Abbe, Emmanuel, Hazla, Jan, and Nachum, Ido
- Subjects
FOS: Computer and information sciences ,polarization ,reed-muller codes ,Computer Science::Emerging Technologies ,capacity ,Information Theory (cs.IT) ,Computer Science - Information Theory ,Computer Science::General Literature ,Data_CODINGANDINFORMATIONTHEORY ,Hardware_LOGICDESIGN ,Computer Science::Information Theory - 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 approach 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.
- Published
- 2020