7 problems

Let $\alpha=(1+\sqrt{5})/2$ be the golden ratio, let $\bf ftm$ be the Fibonacci-Thue-Morse sequence, and let $M_{\bf ftm}(n)$ denote its maximum order complexity at length $n$. Jam…

Let ${\cal S}=(s_i)_{i\geq 0}$ be a binary sequence. Its $N$th maximum order complexity is denoted by $M({\cal S},N)$, and its $N$th expansion complexity by $E_N=E_N({\cal S})$, de…

Let ${\cal S}=(s_i)_{i\geq 0}$ be a binary sequence. Its $N$th maximum order complexity is denoted by $M({\cal S},N)$, and its $N$th expansion complexity by $E({\cal S},N)$, where…

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 $p$ be any odd prime, let $n$ be a positive integer, and set $N=p^n-1$. Let $\{s_t\}_{t=0}^{N-1}$ be the LSB sequence of a $p$-ary $m$-sequence of order $n$, and let $C_p$ deno…

Let $q$ be a prime power, let $S$ be a sequence over the alphabet of size $q$, and let $N_n^{(q-1)}(S)$ denote its maximum-order complexity of order $q-1$ at length $n$. Write…

Let $I(a;\mathcal{F})$ denote the unpredictability of a binary sequence $a\in\{0,1\}^{\infty}$ relative to a predictor class $\mathcal{F}$, and suppose that $\mathcal{F}$ contains…