# Practical Polar Code Construction Using Generalised Generator Matrices

## TL;DR

This paper introduces a method for constructing polar codes with arbitrary lengths using generalized generator matrices, expanding their practical applicability beyond power-of-two lengths.

## Contribution

It proposes a new approach to polar code construction with generalized generator matrices, enabling flexible code lengths and improved performance analysis.

## Key findings

- Generalized generator matrices can produce polar codes of arbitrary lengths.
- The proposed codes achieve similar performance to original polar codes.
- A new polarisation distance measure helps compare different generator matrices.

## Abstract

Polar coding is a recently proposed coding technique that can provably achieve the channel capacity. The polar code structure, which is based on the original 2x2 generator matrix, polarises the channels, i.e., a portion of the channel capacities approach 1, while the remaining channel capacities approach 0. Due to the specific size of this original generator matrix, polar codes can only have code lengths equal to the powers of 2, resulting in inefficiency for codes of practical lengths. In this paper, the performance of finite-length polar codes over the binary erasure channel is analysed. A normalised polarisation distance measure is defined and polar codes from different generator matrices showing different amount of polarisation are compared using this measure. Encoding structures for these generalised polar codes are proposed and polarisation performances in both asymptotical and finite-length cases are investigated for generator matrices of size 3x3 and 4x4. A generalised decoder is also proposed for this generator matrix and its erasure rate is compared with that of the original generator matrix. It is shown that polar codes that have performance similar to the original construction can be constructed and used for a variety of code lengths, not necessarily equal to powers of 2, using generalised generator matrices.