121 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…

Archdeacon's conjecture. An integer $\mathrm{H}(m,n;s;k)$ exists if and only if

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

A Butson Hadamard matrix $BH(n,q)$ is an $n\times n$ complex matrix whose entries are $q$th roots of unity and whose rows are pairwise orthogonal under the standard complex inner p…

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…

Let a simple $\operatorname{T}(t,k,v)$ trade be a trade on a $v$-element set in which no block is repeated. A $(t,k,v)$-halving is a simple $\operatorname{T}(t,k,v)$ trade with vol…

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…

A $\Gamma$-magic rectangle set $\operatorname{MRS}_{\Gamma}(a,b;c)$ on an Abelian group $\Gamma$ of order $abc$ is a collection of $c$ arrays of size $a\times b$, whose entries are…

Let $a,b>1$, let $(\Gamma,+)$ be an abelian group, and let $\Upsilon$ denote the set of all finite abelian groups of odd order or containing more than one involution. An…

Ionin–Shrikhande conjecture. For any Ryser design on $v$ points,

A covering array $\operatorname{CA}(N,k,v)$ is an array with $N$ rows, $k$ columns, and entries from an alphabet of size $v$, such that every choice of two columns contains every o…

Let $m$ be an odd integer with $m>7$ and $m\not\equiv 0 \pmod{3}$. In the additive group $({\mathbb F}_{3^m},+)$, let ${\rm P}$, ${\rm DY}(1)$, ${\rm DY}(-1)$, ${\rm RT}(1)$,…

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…

Exponent reduction conjecture. The exponent $\frac{n(n+1)}{2}-1$ can be reduced to $\frac{n(n-1)}{2}$.

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…

Cyclic Hadamard difference-set conjecture. No cyclic Hadamard difference sets exist other than those in these three families.

Ryser's conjecture. No cyclic difference set exists with

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…

Let $\tau(N)$ be the largest integer $t$ for which an $N$-point three-dimensional spherical $t$-design exists. Since every $t$-design is also a $t'$-design for $t'\leq t$, an $N$-p…

A spherical $t$-design is a finite set of points on the unit sphere whose averages agree with spherical averages for polynomials of degree at most $t$. Let $N$ denote the number of…

A spherical $t$-design is a finite set of points on the unit sphere whose averages agree with spherical averages for polynomials of degree at most $t$. Let $N$ denote the number of…

Golay-number conjecture. The 2-phase Golay numbers are the only possible lengths for 2-phase GCSs.

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 $\mathcal{L}_n$ be the set of Latin squares of even order $n$. In each of the row, column, and symbol views, a two-line trade is obtained by swapping two lines on the support o…

Let $q>r\geq 3$ be integers. An $r$-graph is an $r$-uniform hypergraph, and it is $K_q^r$-divisible when it satisfies the divisibility conditions necessary for a decomposition into…