8 problems

Existence conjecture for weighing matrices. $W(4l,k)$ is nonempty for every $l$ and every $k\le 4l$. This extends the Hadamard-matrix existence problem to weighing matrices; the so…

Let $H$ be a Butson-type matrix, let $k$ be an integer for which the $k$-th exterior product $\wedge^k H$ is defined, and let a generalised weighing matrix mean a matrix satisfying…

For positive integers $m,n,k$, let $W(n,k)$ denote the set of $n\times n$ matrices with entries in $\{-1,0,1\}$ satisfying $WW^\top=kI$. Weighing-matrix extension conjecture. The s…

Existence conjecture. For every positive integer $n$ and every integer $k$ with $1\leq k\leq 4n$, there exists a weighing matrix $W(4n,k)$.

A negacyclic C-matrix is a negacyclic matrix with entries in $\{0,\pm1\}$ satisfying the C-matrix weighing condition. Negacyclic C-matrix nonexistence conjecture. There are no nega…

Nonexistence conjecture for $W(23,16)$. A weighing matrix $W(23,16)$ does not exist.

Goethals–Seidel conjecture. For every positive integer $t$, there exists a weighing matrix

Let ${\cal W}$ be a weighing matrix of order $n$ and weight $w$, and let ${\mathbb C}^{\cal W}_s$ be the associated code; in the comparison under discussion, $s$ is the relevant mo…