Non-Binary Polar Codes using Reed-Solomon Codes and Algebraic Geometry Codes

TL;DR

This paper explores non-binary polar codes constructed from Reed-Solomon and algebraic geometry codes, demonstrating improved performance over binary polar codes and discussing their theoretical properties.

Contribution

It introduces a new construction of non-binary polar codes based on Reed-Solomon and Hermitian codes, with empirical performance analysis and algebraic geometry interpretation.

Findings

4-ary polar codes outperform binary polar codes on AWGN channels

Polar codes using Hermitian codes have asymptotically good performance

Numerical simulations confirm error probability improvements

Abstract

Polar codes, introduced by Arikan, achieve symmetric capacity of any discrete memoryless channels under low encoding and decoding complexity. Recently, non-binary polar codes have been investigated. In this paper, we calculate error probability of non-binary polar codes constructed on the basis of Reed-Solomon matrices by numerical simulations. It is confirmed that 4-ary polar codes have significantly better performance than binary polar codes on binary-input AWGN channel. We also discuss an interpretation of polar codes in terms of algebraic geometry codes, and further show that polar codes using Hermitian codes have asymptotically good performance.