Strange Attractor in Density Evolution

TL;DR

This paper introduces a novel geometric pattern called a strange attractor in the density evolution of polar codes, revealing complex behaviors and new achievable rates for finite block lengths.

Contribution

It defines a strange attractor for synthetic channels in polar coding, linking geometric properties to code construction and complexity reduction techniques.

Findings

Attractor set size equals Fibonacci number for block length N=2^n.

Universal operators increase the set of less reliable synthetic channels.

New achievable rates for finite-length polar codes.

Abstract

The strange attractor represents a complex pattern of behavior in dynamic systems. This paper introduces a strange attractor for synthetic channels in polar coding as a result of a geometric property of density evolution that is a polar code construction technique. First, we define a subset of synthetic channels that are universally less reliable than the original channel. Here, the cardinality of the attractor set is (n+2)-th Fibonacci number for the block length N = 2n. This can be seen as a significantly large number for very long codes. On the other hand, strange attractor can provide new achievable rates for the finite block lengths. Secondly, it is known that polar codes can be constructed with sub-linear complexity by the use of partial orderings. In this study, we additionally define 1 + log2(log2 N) universal operators to reduce the complexity. Then, these universal operators can be applied on the attractor set to increase the number of synthetic channels that are universally less reliable than the natural channel.