15 problems

Let $\mathbb{Z}_n$ denote the cyclic group of order $n$. A circular Costas sequence is equivalent to an injective map $\varphi:\mathbb{Z}_n\to\mathbb{Z}_{n+1}$ whose difference map…

Let $p$ be prime and let a standard circular Costas map be a circular Costas map with domain $\mathbb{Z}_{p^2-1}$ and codomain $\mathbb{Z}_p^2$. Let $\phi$ denote Euler's totient f…

Let $G_1$ and $G_2$ be groups, and let a circular Costas map be a map $\varphi:G_1\to G_2$ that is injective and whose nonzero difference maps are injective; call it standard when…

Let a Costas sequence be a sequence whose difference triangle has no repeated entry in any row. A sequence is singly periodical when every circular shift of its terms is also a Cos…

Higher-dimensional periodic Costas array conjecture. If $A$ is periodic Costas, then $n=2^k$. In particular,

Let $F(x)$ denote the number of primes $p\leq x$ that have at least one Fibonacci primitive root, meaning an element $g$ satisfying $g^2\equiv g+1\pmod p$ and generating the multip…

Elementary-abelian existence conjecture. There are no multidimensional periodic Costas arrays other than multidimensional periodic Costas arrays over the elementary abelian group…

G1–G2 symmetry conjecture. The generators $\mathbf{G1}$ and $\mathbf{G2}$ characterize the full group of symmetries of the multidimensional periodic Welch Costas arrays over…

G1–G2 generation conjecture. These two generators characterize the group of symmetries of the generic $(m+1)$-dimensional periodic Costas arrays over $(\mathbb{Z}_p)^m$.

A honeycomb array is a Costas array with the extra property that every diagonal along the bottom-left--top-right direction contains exactly one dot. The source states that only 12…

A Golomb ruler is a set of marks on a line whose pairwise distances are all distinct. Let the length of the ruler be fixed, and consider the maximum number of marks it can contain.…

Let $k$ be the smallest integer with $k\leq n$ such that there exist $i_1<i_2<\ldots<i_k\in [n]$ for which, for every collection of values $f_1,\ldots,f_k\in[n]$, the set of functi…

A Costas array of order $n$ is a permutation matrix of size $n\times n$ whose displacement vectors between pairs of marked cells are all distinct. A Costas latin square of order…

W1–G2 cross-correlation conjecture.

G2 cross-correlation conjecture.