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

## Abstract

*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. 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 $O(k \log(N) \log^2(k) (\log k + \log\log N))$ to $O(k \log(N) \cdot \log^3(k))$. 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.

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**Submitted**: 29 October 2012

**Accepted**: 15 July 2013

**Published online**: 1 October 2013

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