Capacity-Approaching Polar Codes with Long Codewords and Successive Cancellation Decoding Based on Improved Gaussian Approximation

TL;DR

This paper introduces an improved Gaussian approximation method for constructing long polar codes with successive cancellation decoding over AWGN channels, enabling capacity-approaching performance with reduced computational effort.

Contribution

It proposes an enhanced GA technique based on analyzing the nonlinear function's asymptotic behavior, improving accuracy for large code lengths.

Findings

Successfully designed capacity-approaching polar codes up to length 2^18

Demonstrated the effectiveness of the improved GA in code construction

Achieved low-complexity, high-precision code design for long codewords

Abstract

This paper focuses on an improved Gaussian approximation (GA) based construction of polar codes with successive cancellation (SC) decoding over an additive white Gaussian noise (AWGN) channel. Arikan has proven that polar codes with low-complexity SC decoder can approach the channel capacity of an arbitrary symmetric binary-input discrete memoryless channel, provided that the code length is chosen large enough. Nevertheless, how to construct such codes over an AWGN channel with low computational effort has been an open problem. Compared to density evolution, the GA is known as a low complexity yet powerful technique that traces the evolution of the mean log likelihood ratio (LLR) value by iterating a nonlinear function. Therefore, its high-precision numerical evaluation is critical as the code length increases. In this work, by analyzing the asymptotic behavior of this nonlinear function, we propose an improved GA approach that makes an accurate trace of mean LLR evolution feasible. With this improved GA, through numerical analysis and simulations with code lengths up to $N=2^{18}$, we explicitly demonstrate that various code-rate polar codes with long codeword and capacity approaching behavior can be easily designed.