35 problems

Let ${\bf u}=(u_n)_{n\in\mathbb{N}}$ be a binary sequence that is not eventually periodic. Suppose that the set of return lengths $\mathcal{R}(\mathbf{u})$ is finite, and that … is…

Omega linear-complexity conjecture.

Nonexistence conjecture. No Barker sequences of length greater than $13$ exist.

Schmidt's conjecture. There do not exist perfect binary sequences with

Nonexistence conjecture. There are no perfect binary sequences of period $N>4$.

A binary sequence is a finite sequence of signs, and its merit factor is the quantity … where … with $\epsilon_j\in\{-1,1\}$ and $z\in S^1$. Turyn–Golay's conjecture. The merit fac…

Positive-density reciprocal conjecture. If $A$ contains $0$, is not periodic, and is uniformly distributed modulo every power of $2$, then $\overline{A}$ has positive density. The…

Let $a_1,\ldots,a_n\in\{0,1\}$ and define … The binary sequence $a_1\ldots a_n$ is very odd if every relevant $A_k$ is odd. Pelikán's conjecture. Very odd sequences of length…

Expected exponential linear complexity symmetry gap conjecture. The difference $E_N^{lin-exp}-E_N^{lin-exp-sym}$ is of order of magnitude $2^{N/2}$.

Expected rational complexity symmetry gap conjecture. The difference $E_N^{rat}-E_N^{rat-sym}$ is at least of order of magnitude $2^{N/2}$.

Let $n$ be a positive integer, let $c\geq \left\lceil\frac{n}{2}\right\rceil$, and let $d=n-c$. Let $\mathcal{B}(n,c,d)$ denote the specified class of binary sequences, let…

Let $N_\tau$ denote the number of occurrences of a binary pattern (word) $\tau$ in a sequence. For sequences of length up to $k$, let $A(k)$ be the number of algebraically independ…

Expansion-complexity conjecture for the four sequences. The numerical data suggests that all four sequences have expansion complexity of order of magnitude $N^{1/2}$, with the rest…

Let $\mathcal{M}$ be a binary $m$-sequence of period $2^n-1$, where $n\in\mathbb{N}$. For an integer shift $\tau$ with $1\leq\tau<2^n-1$, let $\mathcal{A}^{A}_{\mathcal{M}}(\tau)$…

Let $h_n$ be the Hankel transform of the Rueppel sequence $r_n$, and let $H_n$ be the Hankel transform of the once-shifted Rueppel sequence $r_{n+1}$. A268411 product conjecture. T…

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform be the sequence of determinants of successive Hankel matrices. A005811 Hankel-transform…

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform be the sequence of determinants of successive Hankel matrices. A005811 Hankel-transform…

Bernasconi's conjecture. Stochastic search procedures will not yield merit factors higher than $5$ for long sequences.

Stochastic-search merit-factor conjecture. Stochastic search procedures will not yield merit factors higher than $5$ for long binary sequences.

Borwein's conjecture. Binary Golay complementary pairs are available only for lengths of the form $2^a 10^b 26^c$, where $a$, $b$, and $c$ are non-negative integers.

Dymacek–Bailey conjecture. There is only one regular Steinhaus graph of odd degree, the complete graph

Uniqueness conjecture. The only perfect sequence is the sequence with $N=4$ and $S=(0,0,0,1,\ldots)$, up to cyclic-shift equivalence.

Let $a(n)$ be defined by writing $n$ in binary, appending a $1$ to each run of $1$'s and a $0$ to each run of $0$'s, and interpreting the resulting binary string as an integer. A r…

For a binary sequence $X$ of length $n$, write $PComS(n,q,c)$ for the family used in the paper, and let a perfect binary sequence have two-level nontrivial autocorrelation with val…

A binary sequence is a sequence with entries in $\{+1,-1\}$. Its autocorrelation merit factor is the reciprocal of its autocorrelation demerit factor, and the asymptotic merit fact…