Capacity Achieving Codes From Randomness Condensers

TL;DR

This paper introduces a framework for constructing small ensembles of capacity-achieving linear codes for various discrete symmetric channels using randomness extractors and condensers, enabling efficient encoding and decoding.

Contribution

It presents a novel general framework leveraging randomness condensers to construct explicit capacity-achieving codes for diverse channels.

Findings

Ensembles maintain capacity-achieving properties under basis change.

Explicit constructions yield polynomial-sized ensembles.

Codes enable near-linear encoding and quadratic decoding times.

Abstract

We establish 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. Using known explicit constructions of condensers, we obtain specific ensembles whose size is as small as polynomial 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.