List Decoding: Algorithms and Applications
TCS '00 Proceedings of the International Conference IFIP on Theoretical Computer Science, Exploring New Frontiers of Theoretical Informatics
Noise-tolerant learning, the parity problem, and the statistical query model
Journal of the ACM (JACM)
Code bounds for multiple packings over a nonbinary finite alphabet
Problems of Information Transmission
On lattices, learning with errors, random linear codes, and cryptography
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
New Results for Learning Noisy Parities and Halfspaces
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
List-decoding reed-muller codes over small fields
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
On the convexity of one coding-theory function
Problems of Information Transmission
A lower bound on list size for list decoding
IEEE Transactions on Information Theory
Combinatorial bounds for list decoding
IEEE Transactions on Information Theory
Error-correcting codes for list decoding
IEEE Transactions on Information Theory
Class of constructive asymptotically good algebraic codes
IEEE Transactions on Information Theory
Justesen's construction--The low-rate case (Corresp.)
IEEE Transactions on Information Theory
On the List-Decodability of Random Linear Codes
IEEE Transactions on Information Theory
Limits to List Decoding of Random Codes
IEEE Transactions on Information Theory
Pseudorandomness
Hi-index | 0.00 |
It is well known that a random q-ary code of rate Ω(ε2) is list decodable up to radius (1 - 1/q - ε) with list sizes on the order of 1/ε2, with probability 1 - o(1). However, until recently, a similar statement about random linear codes has until remained elusive. In a recent paper, Cheraghchi, Guruswami, and Velingker show a connection between list decodability of random linear codes and the Restricted Isometry Property from compressed sensing, and use this connection to prove that a random linear code of rate Ω( ε2 /log3(1/ε)) achieves the list decoding properties above, with constant probability. We improve on their result to show that in fact we may take the rate to be Ω(ε2), which is optimal, and further that the success probability is 1 - o(1), rather than constant. As an added benefit, our proof is relatively simple. Finally, we extend our methods to more general ensembles of linear codes. As an example, we show that randomly punctured Reed-Muller codes have the same list decoding properties as the original codes, even when the rate is improved to a constant.