Pseudo-random generation from one-way functions
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Extractors and pseudorandom generators
Journal of the ACM (JACM)
Randomness conductors and constant-degree lossless expanders
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Introduction to Coding Theory
Extracting all the randomness and reducing the error in Trevisan's extractors
Journal of Computer and System Sciences - STOC 1999
Introduction to Coding Theory
IEEE/ACM Transactions on Networking (TON) - Special issue on networking and information theory
Modern Coding Theory
Linear-time encodable and decodable error-correcting codes
IEEE Transactions on Information Theory - Part 1
Linear-time encodable/decodable codes with near-optimal rate
IEEE Transactions on Information Theory
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We give a general framework for construction of small ensembles of capacity achieving linear codes for a wide range of (not necessarily memoryless) discrete symmetric channels, and in particular, the binary erasure and symmetric channels. The main tool used in our constructions is the notion of randomness extractors and lossless condensers that are regarded as central tools in theoretical computer science. Same as random codes, the resulting ensembles preserve their capacity achieving properties under any change of basis. Our methods can potentially lead to polynomial-sized ensembles; however, using known explicit constructions of randomness conductors we obtain specific ensembles whose size is as small as quasipolynomial in the block length. By applying our construction to Justesen's concatenation scheme (Justesen, 1972) we obtain explicit capacity achieving codes for BEC (resp., BSC) with almost linear time encoding and almost linear time (resp., quadratic time) decoding and exponentially small error probability. The explicit code for BEC is defined and capacity achieving for every block length.