High Rate Multivariate Polynomial Evaluation Codes
Swastik Kopparty, Mrinal Kumar, Harry Sha

TL;DR
This paper introduces new multivariate polynomial evaluation codes that overcome the traditional rate limitations of Reed-Muller codes, achieving high rate and distance with efficient decoding and local testability.
Contribution
The authors construct explicit evaluation domains for multivariate polynomials that yield high-rate codes with good distance and local testability, surpassing classical Reed-Muller rate constraints.
Findings
Codes have relative distance Ω(1) and rate approaching 1 for fixed m.
Two different constructions with efficient decoding algorithms.
One construction exhibits strong locality and local testability.
Abstract
The classical Reed-Muller codes over a finite field are based on evaluations of -variate polynomials of degree at most over a product set , for some less than . Because of their good distance properties, as well as the ubiquity and expressive power of polynomials, these codes have played an influential role in coding theory and complexity theory. This is especially so in the setting of being where they possess deep locality properties. However, these Reed-Muller codes have a significant limitation in terms of the rate achievable -- the rate cannot be more than . In this work, we give the first constructions of multivariate polynomial evaluation codes which overcome the rate limitation -- concretely, we give explicit evaluation domains on which evaluating -variate…
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Taxonomy
TopicsAdvanced Wireless Communication Techniques · Digital Filter Design and Implementation · Coding theory and cryptography
