Design of Polar Code Lattices of Small Dimension

TL;DR

This paper introduces a design method for finite-dimensional polar code lattices that optimizes code rates based on finite-length properties, achieving near-optimal performance with lower complexity.

Contribution

It proposes a novel design technique for polar code lattices using finite-length properties and density evolution, improving performance and complexity over previous methods.

Findings

Achieves a VNR of 2.5 dB at 10^{-4} error rate for dimension 128.

Outperforms BCH code lattices in complexity while maintaining similar performance.

Uses density evolution for rate selection under successive cancellation decoding.

Abstract

Polar code lattices are formed from binary polar codes using Construction D. In this paper, we propose a design technique for finite-dimension polar code lattices. The dimension $n$ and target probability of decoding error are parameters for this design. To select the rates of the Construction D component codes, rather than using the capacity as in past work, we use the explicit finite-length properties of the polar code. Under successive cancellation decoding, density evolution allows choosing code rates that satisfy the equal error probability rule. At an error-rate of $10^{-4}$, a dimension $n=128$ polar code lattice achieves a VNR of 2.5 dB, within 0.2 dB of the best-known BCH code lattice, but with significantly lower decoding complexity.