Capacity Achieving Codes From Randomness Conductors

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.