# Extending de Bruijn sequences to larger alphabets

## TL;DR

This paper introduces a method to extend de Bruijn sequences to larger alphabets by embedding existing sequences into higher-order sequences with additional symbols, ensuring no long runs without the new symbol.

## Contribution

It presents a novel approach using auxiliary graphs and maximum flow algorithms to extend de Bruijn sequences to larger alphabets.

## Key findings

- Successful embedding of de Bruijn sequences into larger alphabets.
- Ensured no long runs without the new symbol in extended sequences.
- Method applicable to various alphabet sizes and sequence orders.

## Abstract

A de Bruijn sequence of order n over a k-symbol alphabet is a circular sequence where each length-n sequence occurs exactly once. We present a way of extending de Bruijn sequences by adding a new symbol to the alphabet: the extension is performed by embedding a given de Bruijn sequence into another one of the same order, but over the alphabet with one more symbol, while ensuring that there are no long runs without the new symbol. Our solution is based on auxiliary graphs derived from the de Bruijn graph and solving a problem of maximum flow.