12 problems

Let $n,t\geq 2$ and let $w<t$. Let $\mathcal{V}_t(n,w)$ be the universal cycle constructed from the relevant fixed-weight strings, and let $\mathcal{V'}_t(n,w)$ be the sequence obt…

Let $n\geq 3$, let $c$ be an integer satisfying … and suppose $c\neq 2^{n-1}+n+1$. A span $n$ de Bruijn sequence is a cyclic binary sequence in which every binary word of length…

PRR feedback-function conjecture. The pure run-length register is conjectured to be the underlying feedback function used in a lexicographic composition construction.

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

Let $s$ be a de Bruijn sequence over an alphabet of size $q$, and for each letter $a$ let $f_a(i)$ denote its frequency in the initial subsequence of length $i$. Define the discrep…

Let $\mathbf{s}$ be a modified binary de Bruijn sequence of order $n$, and let $\Delta_{\mathbf{s}}$ denote its linear span. Tan's conjecture. … This conjecture concerns the possib…

Let $alt(n)$ denote the alternating sequence of $0$s and $1$s of length $n$ that ends with $0$. Let $\d835\d835\mathcal{L}_n$ be the de Bruijn sequence obtained from the lexicograp…

Maximum discrepancy conjecture. The sequence $\mathcal{DB}_{max}(n)$ has discrepancy equal to

Quadratic discrepancy conjecture. The de Bruijn sequence constructed by the Pref-opposite algorithm has discrepancy $\Theta(n^2)$.

Let $F_h$ be the feedback function defined from a nonsingular function $h$ in Equation, with $n\geq m+2$. An input pair consists of a rooted spanning tree and an initial state, and…

Chang's conjecture. Any two binary de Bruijn sequences obtained from maximal-period linear feedback shift register sequences of the same order $k$ have the same number of cross-joi…

Discrepancy asymptotic constant conjecture. There is some constant $C$ such that