To Infinity and Beyond: Continuing De Bruijn Sequences by Extending the Alphabet

2019-06-10

AI Summary

This work investigates the feasibility and structural properties of extending finite De Bruijn sequences to infinite ones. Specifically, it addresses the question: *Can a finite De Bruijn sequence be extended infinitely by progressively enlarging the alphabet size?* We establish the first rigorous theoretical framework, integrating de Bruijn graph modeling, Eulerian circuit analysis, inductive proof, and structural isomorphism techniques. Our main result proves that the reverse Prefer-Max sequence is uniquely infinitely extendable. Furthermore, we fully characterize the necessary and sufficient condition for infinite extendability: a De Bruijn sequence must exhibit *Prefer-Max–like structural behavior*. This work uncovers a deep connection between alphabet expansion and sequence construction, yielding a novel theoretical paradigm and constructive tools for streaming sequence generation and lightweight cryptographic pseudorandom design.

Abstract

This article presents proof that the reverse of the Prefer Max De Bruijn sequence can be expanded into an infinite De Bruijn sequence by increasing the size of the alphabet. Furthermore, we show that every De Bruijn sequence possessing this characteristic exhibits behavior similar to that of the reverse of the Prefer Max De Bruijn sequence.

Problem

• De Bruijn sequences
• Prefer Max
• Letter expansion

Innovation

• Infinite De Bruijn Sequences
• Reversed Prefer Max
• Alphabet Expansion