Using concatenated algebraic geometry codes in channel polarization

TL;DR

This paper investigates the use of concatenated algebraic geometry codes in binary polar codes, demonstrating optimal code choices for different kernel sizes to improve error correction performance.

Contribution

It introduces a method of using concatenated algebraic geometry codes in binary polar codes and compares their performance across various kernel sizes.

Findings

Concatenated Reed-Solomon codes outperform others for kernels up to size 1800.

For larger kernels, Hermitian or Suzuki codes perform better.

Optimal code choice depends on kernel size and field characteristics.

Abstract

Polar codes were introduced by Arikan in 2008 and are the first family of error-correcting codes achieving the symmetric capacity of an arbitrary binary-input discrete memoryless channel under low complexity encoding and using an efficient successive cancellation decoding strategy. Recently, non-binary polar codes have been studied, in which one can use different algebraic geometry codes to achieve better error decoding probability. In this paper, we study the performance of binary polar codes that are obtained from non-binary algebraic geometry codes using concatenation. For binary polar codes (i.e. binary kernels) of a given length $n$, we compare numerically the use of short algebraic geometry codes over large fields versus long algebraic geometry codes over small fields. We find that for each $n$ there is an optimal choice. For binary kernels of size up to $n \leq 1,800$ a concatenated Reed-Solomon code outperforms other choices. For larger kernel sizes concatenated Hermitian codes or Suzuki codes will do better.