38 problems

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…

Let $n>q>r\geq 1$ be integers. An $(n,q,r)$-Steiner system is a collection of $q$-subsets of an $n$-element set such that every $r$-subset is contained in exactly one member. Equiv…

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…

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

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 $(h_1\dots h_k)$ be a partition with $h_1\geq\dots\geq h_k>0$, and let an RP be a realization of a partition by disjoint subsquares of a Latin square. Colbourn's conjecture. If…

Let $(h_1h_2\dots h_k)$ be a partition with $h_1\geq h_2\geq\dots\geq h_k>0$, and let an RP be a realization of a partition by disjoint subsquares of a Latin square. Colbourn's con…

Simplex design factorial-size conjecture. There exists a $t$-design $X\subset \mathbb{R}^d/\left(\sum_{i=1}^d x_i=-1\right)$ for $\sigma$, with $t\leq d$, such that

Existence conjecture. $(r \times c, v)$-near triple arrays exist for any $c \geq r(r - 1)$ and $v \geq c$.

Let $G$ be an abelian group of odd order $g\geq 3$. A strong starter in $G$ is a starter whose pair sums are distinct and nonzero. Horton's conjecture. There is a strong starter in…

Let $n\geq 5$ be an odd integer. A strong starter in the cyclic group $\mathbb{Z}_n$ is a starter satisfying the additional distinct nonzero pair-sum condition defined in the paper…

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

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 $h_1,\dots,h_5$ be positive integers, let $n=h_1+\cdots+h_5$, and let $[5]=\{1,2,3,4,5\}$. A $\operatorname{2-RP}(h_1\dots h_5)$ is a two-repetition gerechte design whose five…

Let $m>3$, and consider the voltage group $\mathbb Z_m$. For parameters $a,b$, define the two parameter sets $F_1(m;a,b)$ and $F_2(m;a,b)$ by … for $a\geq b$ and…

Existence conjecture for symmetric $[\alpha]$-Hadamard matrices. For every $\alpha\geq 3$, there exist symmetric $[\alpha]$-Hadamard matrices of all orders.

Existence conjecture. For each D-optimal parameter set $(v;r,s;\lambda)$ there exists a difference family over some abelian group of order $v$ having these parameters.

Let $(\Gamma,+)$ be a finite abelian group, and let $\mathcal{G}$ be the set of finite abelian groups that either have odd order or contain more than one involution. For integers…

For integers $n$ and $k$, let $g(n,k)$ be the minimum covering multiplicity for a perfect sequence covering array with parameters $(n,k)$. The previously known exceptional values i…

Let $V$ be a set of $v$ servers, and let a data placement be a family of $b$ blocks, each of size $k$, on $V$. For a placement $(V,\mathcal{F})$, let … where $v_j$ is the number of…

Symmetry-symbol conjecture. For each normalized and non-exceptional PPS $(v;k_1,k_2,k_3,k_4;\lambda)$ there exist PDFs with symmetry symbols $(s{**})$ and $({**}s)$.

Type 2 existence conjecture. Legendre difference families of type $2$ exist for all odd lengths $v>8$.

Let $n$ be an integer with $n\geq11$, and let $\mathbb{Z}_n$ be the additive cyclic group of order $n$. A strong Skolem starter is a starter for $\mathbb{Z}_n$ that is both Skolem…

Pasotti–Pellegrini conjecture. There exists a near $1$-factor $F$ of $K_v$ with $\ell(F)=L$ if and only if $L$ is admissible.