113 problems

Hadamard's conjecture. Such a matrix exists whenever $4\mid n$. This is a central existence problem in design theory. The paper notes that order $668$ is the smallest order present…

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…

Let $H\in\{\pm1\}^{n\times n}$ be a circulant Hadamard matrix, equivalently a perfect binary sequence of length $n$. The length-$4$ sequence $[1,-1,-1,-1]$ is the only nontrivial p…

Paley's conjecture. There is a binary Hadamard matrix of every order divisible by $4$.

A partial complex Hadamard matrix $H\in M_{M\times N}(\mathbb T)$ is regular when the scalar products between its rows decompose as sums of cycles. A Butson matrix is a complex Had…

Let $BS(m,n)$ denote four sequences $A,B,C,D$ of $71$ of lengths $m,m,n,n$ with zero combined non-periodic autocorrelation, and let $BS(n+1,n)$ be the corresponding case. Base sequ…

Even Williamson conjecture. Williamson matrices exist for every even order $n$.

Let $BS(n+1,n)$ be the set of base sequences of the indicated lengths. A base sequence $(A;B;C;D)\in BS(n+1,n)$ is near-normal when $a_i=(-1)^{i-1}b_i$ for every $i$ with…

An almost Hadamard matrix in the complex sense is the notion introduced in the cited work, while a complex Hadamard matrix is a unitary matrix whose entries have equal modulus afte…

A base sequence of type $BS(r,s)$ is a quadruple of sequences $(A;B;C;D)$ with the standard base-sequence conditions, and $BS(r,s)$ denotes the set of all such quadruples. The nota…

Let $n$ be a positive integer. A Bush-type Hadamard matrix of order $4n^2$ is a Hadamard matrix whose entries are $2n\times 2n$ blocks and whose diagonal blocks are $J_{2n}$, while…

Let $R_n$ be an $n\times n$ matrix with entries in $\{-1,1\}$. Hadamard's bound gives … and equality can occur only when $n=1$, $2$, or $n\equiv0\pmod 4$. Paley's conjecture. Equal…

Orrick–Solomon's conjecture. The determinant $72\times9^{17}\times2^{36}$ achieves the global maximum among determinants of $37\times37$ sign matrices.

Let $\operatorname{md}(n)$ denote the maximum determinant among all $n\times n$ matrices whose entries are $\pm1$. The authors exhibit a matrix of order $22$ with determinant…

For fixed $N$, let $B_{N-1,K}$ and $W_N^{\rm opt}$ denote the two-branch and standard dyadic Walsh--Hadamard truncation matrices described in the paper, with $K=0,\dots,N$. Two-bra…

Let $\mathcal{C}$ be a set of columns of a dyadic-truncated Walsh--Hadamard matrix $W_\Phi$ with exactly one node, such that each branch is complete above the node. Let…

Let $H$ be a Kimura Hadamard matrix (KHM) of order $4(2k)+4$, and let $(R,S)\in\operatorname{Aut}(H)$ be an automorphism of prime order $p$. Prime-order divisor conjecture. Then ……

Let $H$ be a Kimura Hadamard matrix (KHM) of order $8k+4$. An automorphism is a pair $(R,S)$ acting on $H$ by the relation $RH=HS$. The known automorphisms have block-diagonal form…

Let $H$ be a Kimura Hadamard matrix (KHM) of order $8k+4$, and let $\operatorname{Aut}(H)$ denote its automorphism group. Ó Catháin's conjecture. The order of the automorphism grou…

A quaternionic Hadamard matrix (QHM) is a quaternionic matrix whose entries have unit norm and whose rows are mutually orthogonal. Two QHMs are equivalent under the equivalence ope…

Skew-Hadamard matrix existence conjecture. The upper bound is saturated if and only if an $n\times n$ skew-Hadamard matrix exists, and such a matrix should exist whenever $n$ is di…

Embedding conjecture. If $H$ is a regular Hadamard matrix of order $v^2$, then such a containing cube and subcube exist.

Let $q>11$ be a prime power with $q\equiv3\bmod4$, let $d=q+1$, and let $H$ be the Paley I Hadamard matrix of order $d$. Write $\tilde H$ for the associated object and…

Let $p$ be an odd prime, and let a $p$-modular Hadamard matrix of order $d$ mean the matrix notion used in the paper. The $p$-modular Hadamard conjecture. For all but finitely many…

Let a SIC be a system of $d^2$ equiangular lines, represented here by vectors over a finite field extension as in the paper. A SIC is totally asymmetric when its automorphism group…