# Two New Families of Quantum Synchronizable Codes

## TL;DR

This paper introduces two novel quantum synchronization coding methods based on specific constructions, expanding the types of available codes and demonstrating improved error correction capabilities over traditional BCH codes.

## Contribution

The paper proposes two new quantum synchronization code families using the $(m{u}+m{v}|m{u}-m{v})$ and product constructions, enhancing code variety and error correction performance.

## Key findings

- Maximum synchronization error tolerance conditions identified.
- Repeated-root cyclic codes outperform BCH codes in Pauli error correction.
- New code families significantly enrich quantum synchronization coding options.

## Abstract

In this paper, we present two new ways of quantum synchronization coding based on the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction and the product construction respectively, and greatly enrich the varieties of available quantum synchronizable codes. The circumstances where the maximum synchronization error tolerance can be reached are explained for both constructions. Furthermore, repeated-root cyclic codes derived from the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction are shown to be able to provide better Pauli error-correcting capability than BCH codes.