Efficient data compression from statistical physics of codes over finite fields

TL;DR

This paper introduces a low-complexity, near-optimal data compression method for binary sources using the cavity method over GF(q), leveraging sparse low density parity check codes and reinforced belief propagation.

Contribution

It presents a novel compression scheme based on GF(q) codes with a geometrical modification of codeword space, enabling efficient compression and decompression.

Findings

Achieves near-optimal empirical performance

Computational complexity is O(d.n.q.log(q)) per iteration

Decompression is linear in code length using leaf-removal

Abstract

In this paper we discuss a novel data compression technique for binary symmetric sources based on the cavity method over a Galois Field of order q (GF(q)). We present a scheme of low complexity and near optimal empirical performance. The compression step is based on a reduction of sparse low density parity check codes over GF(q) and is done through the so called reinforced belief-propagation equations. These reduced codes appear to have a non-trivial geometrical modification of the space of codewords which makes such compression computationally feasible. The computational complexity is O(d.n.q.log(q)) per iteration, where d is the average degree of the check nodes and n is the number of bits. For our code ensemble, decompression can be done in a time linear in the code's length by a simple leaf-removal algorithm.