Subspace Evasive Sets

TL;DR

This paper presents an explicit construction of large subspace-evasive sets over large finite fields, which are used to improve the list-decoding of folded Reed-Solomon codes by reducing the list size to a constant.

Contribution

It provides a new explicit construction of subspace-evasive sets over large fields and applies them to enhance list-decoding of Reed-Solomon codes.

Findings

Constructed subspace-evasive sets over large finite fields.

Reduced list size of folded Reed-Solomon codes from poly(n) to a constant.

Enabled explicit list-decoding with optimal rate and constant list-size.

Abstract

In this work we describe an explicit, simple, construction of large subsets of F^n, where F is a finite field, that have small intersection with every k-dimensional affine subspace. Interest in the explicit construction of such sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl (2004) who showed how such constructions over the binary field can be used to construct explicit Ramsey graphs. More recently, Guruswami (2011) showed that, over large finite fields (of size polynomial in n), subspace evasive sets can be used to obtain explicit list-decodable codes with optimal rate and constant list-size. In this work we construct subspace evasive sets over large fields and use them to reduce the list size of folded Reed-Solomon codes form poly(n) to a constant.