31 problems

Let $\gamma\in(\pm1)^N$ be a binary vector, and define the circulant matrix $H\in M_N(\pm1)$ by … with indices interpreted modulo $N$. A circulant Hadamard matrix has pairwise orth…

Standard-circulant first appearance value conjecture. For $n \geq 2$,

Standard-circulant first appearance degree conjecture. For $n \geq 2$,

Strong $d\times 2d$ conjecture. For each $d\neq4$, there exists a real smooth submanifold

Medium $d\times 2d$ conjecture. For every $d$, there exists a $2$-circulant $d\times 2d$ ETF.

Let $n>4$, let $A$ be an $n\times n$ circulant matrix with entries in $\{-1,1\}$, and let $\|\cdot\|$ denote the Euclidean norm on $\mathbb R^n$. Quantitative Ryser conjecture. The…

Circulant equality conjecture.

For each integer $n\geq 6$, let $D_{n,3}$ denote the single-block matrix used in the paper's circulant block-matrix notation. Hefner et al.'s conjecture. … This is a special case o…

Let $\mathcal{C}_d$ denote the set of circulant matrices, $\mathcal{U}_d$ the set of unistochastic matrices, and $\mathcal{C}_d^2$ the set of doubly circulant unistochastic matrice…

Let $R$ be a commutative finite chain ring (CFCR) of nilpotency index $e$, and let $n$ be a positive integer. Let $C_n(R)$ denote the ring of $n\times n$ circulant matrices over…

Let $R$ be a commutative finite chain ring (CFCR) of nilpotency index $e$, let $n$ be a positive integer, and suppose that the maximal ideal of $R$ is generated by $gamma$. Write…

Odd-parameter existence conjecture. Such a matrix exists only when

Let $p$ be a prime, let $G=SL_2(\mathbb{F}_p)$, and let $f:G\rightarrow\{0,1\}$ be chosen uniformly at random. Write $M_G(f)$ for the corresponding $G$-circulant matrix. Rigidity c…

All-odd-orders conjecture. A circulant good matrix exists in every odd order $n$.

A circulant complex Hadamard matrix is a complex Hadamard matrix whose rows are cyclic shifts of one another. Arasu–de Launey–Ma conjecture. There is no circulant complex Hadamard…

A circulant Hadamard code of length $4t$ is the binary code formed by the rows of a normalized binary circulant Hadamard matrix and their complements. Circulant Hadamard code nonex…

Let $n$ be a positive integer. A Hadamard matrix of order $2n+2$ with two circulant cores is a matrix whose two core blocks $A$ and $B$ are circulant matrices of order $n$ and sati…

Let $p$ be a positive integer. A Hadamard matrix with one circulant core of order $p+1$ has block form $\left(\begin{smallmatrix}1&e\\e^{t}&X_C\end{smallmatrix}\right)$, where…

One-circulant-core order claim. These are the only possible orders for a Hadamard matrix with one circulant core.

Let $A$ be an $n\times n$ circulant matrix whose entries are real numbers in the interval $[0,1]$. The first row determines $A$ through parameters $a_0,\ldots,a_{n-1}$, so these ma…

Let $D_{01}(n)$ denote the maximum determinant of an $n\times n$ circulant matrix with entries in $\{0,1\}$, and let $U_{01}(n)$ be the upper bound defined by … Here…

Generalized circulant Hadamard conjecture. The relation

Generalized minor inequality sufficiency conjecture. Generalized minor inequalities, together with boolean facets and the rank constraint, are sufficient for describing

Let $C_n^{\operatorname{circ}}(\infty)$ denote the set of circulant complex Hadamard matrices of order $n$, with no fixed bound on the orders of their root-of-unity entries, and le…

For $N=4$, decompose the functional into components $\Phi=\Phi_0+\Phi_1+\Phi_2+\Phi_3$, where the $\Phi_i$ are the components defined in the preceding formulas. Parity conjecture f…