The eBugs conjecture on the existence of de Bruijn families

Let $\mathcal{A}$ be an alphabet, and let $m$, $p$, and $n$ be positive integers. A de Bruijn family of order $n$ on $\mathcal{A}$ is an unordered collection of $m$ cyclic strings, each of length $p$, such that every string of length $n$ over $\mathcal{A}$ appears exactly once as a substring of a unique member. Denote the set of such families by $\operatorname{deBF}(\mathcal{A},m,p,n)$.

The eBugs conjecture. There exists a de Bruijn family in $\operatorname{deBF}(\mathcal{A},m,p,n)$ if and only if

$$\nm p=|\mathcal{A}|^n$$

\nand $m>n$.

The counting condition $m p=|\mathcal{A}|^n$ is necessary because the family contains exactly $m p$ cyclic substrings of length $n$. The conjecture asserts that this condition, together with $m>n$, is also sufficient, giving existence for a broad range of parameters motivated by the eBugs robot-identification problem.