# Bit-Level Construction for Multiplicative-Repetition-Based Non-Binary Polar Codes

## TL;DR

This paper introduces a new bit-level construction method for non-binary polar codes that improves decoding performance by optimizing bit allocation in synthesized channels.

## Contribution

The novel bit-level construction method for MR-based non-binary polar codes optimizes bit allocation using error probability and channel capacity.

## Key findings

- Bit-level construction improves decoding performance compared to conventional symbol-level methods.
- A Monte-Carlo method reduces complexity in determining optimal bit allocation.
- Probabilistic shaping further enhances performance in MR-based non-binary polar codes.

## Abstract

In this paper, we discuss non-binary polar codes using a 2×2 matrix over a Galois field GF(2q) as the kernel. Conventional construction of non-binary polar codes divides the synthesized channels into frozen channels and information channels. Each information channel carries one symbol, i.e., q bits. However, there are many middle channels with insufficient polarization, which cannot carry one symbol of q bits but only i bits, 1≤i<q,i∈Z, at finite block length. In this paper, we consider bit-level construction for multiplicative repetition (MR)-based non-binary polar codes and propose a bit-level construction based on the two following methods. We first calculate the error probability and channel capacity lower bound of each synthesized channel based on the channel degradation method, and then determine both the number and index of the carried bits for each synthesized channel according to the symbol error probability and capacity. To reduce complexity, we also introduce a Monte-Carlo method. We compute the error probability of each synthesized channel carrying i information bits and select the optimal construction that can minimize the union bound of the error probability. Finally, an improved construction-based probabilistic shaping method for MR-based non-binary polar codes is considered. Simulation results show that the proposed construction significantly improved the decoding performance compared with the conventional construction scheme.