# Maximal State Complexity and Generalized de Bruijn Words

## TL;DR

This paper determines the maximum state complexity for certain languages, characterizes those achieving it, and introduces a generalized de Bruijn words concept with existence proofs and enumeration.

## Contribution

It provides exact maximum state complexity values, characterizes languages reaching these maxima, and generalizes de Bruijn words with existence and counting results.

## Key findings

- Maximum state complexity for languages of m words of length N computed
- Characterization of languages achieving maximum complexity
- Existence and enumeration of generalized de Bruijn words

## Abstract

We compute the exact maximum state complexity for the language consisting of $m$ words of length $N$, and characterize languages achieving the maximum. We also consider a special case, namely languages $C(w)$ consisting of the conjugates of a single word $w$. The words for which the maximum state complexity of $C(w)$ is achieved turn out to be a natural generalization of de Bruijn words. We show that generalized de Bruijn words exist for each length and consider the number of them.