PAC codes with Bounded-Complexity Sequential Decoding: Pareto Distribution and Code Design

TL;DR

This paper analyzes the decoding complexity of PAC codes, showing that adhering to polarized cutoff rate constraints bounds complexity and improves performance, achieving significant gains over 5G polar and LDPC codes.

Contribution

It demonstrates that PAC codes following polarized cutoff rate constraints have bounded complexity with Pareto distribution, guiding effective code design.

Findings

Decoding complexity can be bounded with Pareto distribution under certain rate-profile constraints.

PAC codes constructed with these constraints outperform 5G polar and LDPC codes at high rates.

Simulation shows a 0.75 dB gain at FER of 10^{-5} for the proposed PAC code.

Abstract

Recently, a novel variation of polar codes known as polarization-adjusted convolutional (PAC) codes has been introduced by Ar{\i}kan. These codes significantly outperform conventional polar and convolutional codes, particularly for short codeword lengths, and are shown to operate very close to the optimal bounds. It has also been shown that if the rate profile of PAC codes does not adhere to certain polarized cutoff rate constraints, the computation complexity for their sequential decoding grows exponentially. In this paper, we address the converse problem, demonstrating that if the rate profile of a PAC code follows the polarized cutoff rate constraints, the required computations for its sequential decoding can be bounded with a distribution that follows a Pareto distribution. This serves as a guideline for the rate-profile design of PAC codes. For a high-rate PAC\,$(1024,899)$ code, simulation results show that the PAC code with Fano decoder, when constructed based on the polarized cutoff rate constraints, achieves a coding gain of more than $0.75$ dB at a frame error rate (FER) of $10^{-5}$ compared to the state-of-the-art 5G polar and LDPC codes.