Concatenated codes can achieve list-decoding capacity
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Better Binary List-Decodable Codes Via Multilevel Concatenation
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Limits to List Decoding Random Codes
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
Better binary list decodable codes via multilevel concatenation
IEEE Transactions on Information Theory
Private Interrogation of Devices via Identification Codes
INDOCRYPT '09 Proceedings of the 10th International Conference on Cryptology in India: Progress in Cryptology
Efficient list decoding of explicit codes with optimal redundancy
AAECC'07 Proceedings of the 17th international conference on Applied algebra, algebraic algorithms and error-correcting codes
The existence of concatenated codes list-decodable up to the hamming bound
IEEE Transactions on Information Theory
Efficiently decodable non-adaptive group testing
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Algorithms and theory of computation handbook
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Fuzzy vault for multiple users
AFRICACRYPT'12 Proceedings of the 5th international conference on Cryptology in Africa
ℓ2/ℓ2-Foreach sparse recovery with low risk
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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Error correcting codes systematically introduce redundancy into data so that the original information can be recovered when parts of the redundant data are corrupted. Error correcting codes are used ubiquitously in communication and data storage. The process of recovering the original information from corrupted data is called decoding. Given the limitations imposed by the amount of redundancy used by the error correcting code, an ideal decoder must efficiently recover from as many errors as information-theoretically possible. In this thesis, we consider two relaxations of the usual decoding procedure: list decoding and property testing. A list decoding algorithm is allowed to output a small list of possibilities for the original information that could result in the given corrupted data. This relaxation allows for efficient correction of significantly more errors than what is possible through usual decoding procedure which is always constrained to output the transmitted information. (1) We present the first explicit error correcting codes along with efficient list-decoding algorithms that can correct a number of errors that approaches the information-theoretic limit. This meets one of the central challenges in the theory of error correcting codes. (2) We also present explicit codes defined over smaller symbols that can correct significantly more errors using efficient list-decoding algorithms than existing codes, while using the same amount of redundancy. (3) We prove that an existing algorithm for a specific code family called Reed-Solomon codes is optimal for "list recovery," a generalization of list decoding. Property testing of error correcting codes entails "spot checking" corrupted data to quickly determine if the data is very corrupted or has few errors. Such spot checkers are closely related to the beautiful theory of Probabilistically Checkable Proofs (or PCPs). (1) We present spot checkers that only access a nearly optimal number of data symbols for an important family of codes called Reed-Muller codes. Our results are the first for certain classes of such codes. (2) We define a generalization of the "usual" testers for error correcting codes by endowing them with the very natural property of "tolerance," which allows slightly corrupted data to pass the test.