Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes

TL;DR

This paper proves that random linear codes over finite fields are highly likely to be efficiently list decodable near the optimal radius, using properties of Fourier matrices and the Restricted Isometry Property, with implications for compressed sensing.

Contribution

It establishes new probabilistic bounds on list decodability of random linear codes and links these bounds to the Restricted Isometry Property of associated matrices, improving understanding of code performance.

Findings

Random linear codes are list decodable at near-optimal radius with high probability.

The analysis connects list decoding guarantees to the Restricted Isometry Property of Fourier-related matrices.

Improved bounds on the number of samples needed for sparse signal reconstruction in compressed sensing.

Abstract

We prove that a random linear code over F_q, with probability arbitrarily close to 1, is list decodable at radius (1-1/q-\epsilon) with list size L=O(1/\epsilon^2) and rate R=\Omega_q(\epsilon^2/(log^3(1/\epsilon))). Up to the polylogarithmic factor in (1/\epsilon) and constant factors depending on q, this matches the lower bound L=\Omega_q(1/\epsilon^2) for the list size and upper bound R=O_q(\epsilon^2) for the rate. Previously only existence (and not abundance) of such codes was known for the special case q=2 (Guruswami, H{\aa}stad, Sudan and Zuckerman, 2002). In order to obtain our result, we employ a relaxed version of the well known Johnson bound on list decoding that translates the average Hamming distance between codewords to list decoding guarantees. We furthermore prove that the desired average-distance guarantees hold for a code provided that a natural complex matrix encoding the codewords satisfies the Restricted Isometry Property with respect to the Euclidean norm (RIP-2). For the case of random binary linear codes, this matrix coincides with a random submatrix of the Hadamard-Walsh transform matrix that is well studied in the compressed sensing literature. Finally, we improve the analysis of Rudelson and Vershynin (2008) on the number of random frequency samples required for exact reconstruction of k-sparse signals of length N. Specifically, we improve the number of samples from O(k log(N) log^2(k) (log k + loglog N)) to O(k log(N) log^3(k)). The proof involves bounding the expected supremum of a related Gaussian process by using an improved analysis of the metric defined by the process. This improvement is crucial for our application in list decoding.