17 problems

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…

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…

For each natural number $n$, let $S(n)$ denote the number of very odd binary sequences of length $n$. For an integer $e\geq 1$, consider the level set of lengths satisfying…

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 ${\bf u}=(u_n)_{n\in\mathbb{N}}$ 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.

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 $r(x)$ 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…

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…

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

Linear-complexity conjecture. If $2^{p-1}\not\equiv 1\pmod {p^2}$, then the linear complexity $L$ of $s^\infty$ is

Complementary Cusick conjecture. For every nonnegative integer $t$,

Tu–Deng conjecture. For every such $k$ and $t$, one has

Let $q$ be an odd prime power, let $\theta$ be primitive in $\mathbb{F}_q$, and let $Z_{n,\theta}$ be the Sidelnikov sequence of length $n=q-1$ with respect to $\theta$. Let $F$ de…

Let $F=\mathbb{F}_{2^d}$ and let $K$ be a subfield of size $2^k$, where $k$ divides $d$. Let $\theta$ be primitive in $F$, let $\ell$ be coprime to $2^k-1$, and let the Gordon-Mill…

Periodicity conjecture. The sequence $\{\gamma_a(n)\}$ is periodic. If $2^m\mathrel{\|}a$, then its minimal period has $2^{m+1}$ terms. If $a$ is evil, this period consists of the…