Finite fields
Decoding of Reed Solomon codes beyond the error-correction bound
Journal of Complexity
Cryptographic Hardness Based on the Decoding of Reed-Solomon Codes
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Hardness of Approximating the Minimum Distance of a Linear Code
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Robust pcps of proximity, shorter pcps and applications to coding
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
On the List and Bounded Distance Decodibility of the Reed-Solomon Codes (Extended Abstract)
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Limits to list decoding Reed-Solomon codes
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Improved decoding of Reed-Solomon and algebraic-geometry codes
IEEE Transactions on Information Theory
Bounds on list decoding of MDS codes
IEEE Transactions on Information Theory
Nonlinear codes from algebraic curves improving the Tsfasman-Vladut-Zink bound
IEEE Transactions on Information Theory
List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
| Hi-index | 754.84 |
We show combinatorial limitations on efficient list decoding of Reed-Solomon codes beyond the Johnson-Guruswami-Sudan bounds. In particular,we showthat for arbitrarily large fields FN, |FN| = N, for any δ Ɛ, (0,1) and K = Nδ : • Existence: there exists a received word wN : FN → FN that agrees with a super-polynomial number of distinct degree K polynomials on ≈ N√δ points each; • Explicit: there exists a polynomial time constructible received word w1N : FN → FN that agrees with a superpolynomial number of distinct degree K polynomials, on ≈ 2√logN K points each. In both cases, our results improve upon the previous state of the art, which was ≈ Nδ/δ points of agreement for the existence case (proved by Justesen and Hoholdt), and ≈ 2Nδ points of agreement for the explicit case (proved by Guruswami and Rudra). Furthermore, for δ close to 1 our bound approaches the Guruswami-Sudan bound (which is √NK) and implies limitations on extending their efficient Reed-Solomon list decoding algorithm to larger decoding radius. Our proof is based on some remarkable properties of subspace polynomials. Using similar ideas, we then present a family of low rate codes that are efficiently list-decodable beyond the Johnson bound. This leads to an optimal list-decoding algorithm for the family of matrixcodes.