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A sequence covering array, denoted \textsf{SCA}$(N;t,v)$, is a set of $N$ permutations of $\{0, \dots, v-1 \}$ such that each sequence of $t$ distinct elements of $\{0, \dots, v-1\}$ reads left to right in at least one permutation. The minimum number
Externí odkaz:
http://arxiv.org/abs/2411.17145
Autor:
Waldron, Shayne
We give a simple presentation of the six quaternionic equiangular lines in $\mathbb{H}^2$ as an orbit of the primitive quaternionic reflection group of order 720 (which is isomorphic to 2.A_6 the double cover of $A_6)$. Other orbits of this group are
Externí odkaz:
http://arxiv.org/abs/2411.16766
Taking inspiration from [1, 21, 24], we develop a general framework to deal with the model theory of open incidence structures. In this first paper we focus on the study of systems of points and lines (rank $2$). This has a number of applications, in
Externí odkaz:
http://arxiv.org/abs/2411.10792
We derive universal lower and upper bounds for max-min and min-max problems (also known as polarization) for the potential of spherical $(k,k)$-designs and provide certain examples, including unit-norm tight frames, that attain these bounds. The univ
Externí odkaz:
http://arxiv.org/abs/2411.00290
We prove a conjecture of Gy\'arf\'as (1976), which asserts that any family of trees $T_1, \dots, T_{n}$ where each $T_k$ has $k$ vertices packs into $K_n$. We do so by translating the decomposition problem into a labeling problem, namely complete lab
Externí odkaz:
http://arxiv.org/abs/2410.13840
Let $m,n,s,k$ be four integers such that $1\leqslant s \leqslant n$, $1\leqslant k\leqslant m$ and $ms=nk$. A signed magic array $SMA(m,n; s,k)$ is an $m\times n$ partially filled array whose entries belong to the subset $\Omega\subset \mathbb{Z}$, w
Externí odkaz:
http://arxiv.org/abs/2410.04101
Autor:
Roucairol, Milo, Cazenave, Tristan
We are interested in the automatic refutation of spectral graph theory conjectures. Most existing works address this problem either with the exhaustive generation of graphs with a limited size or with deep reinforcement learning. Exhaustive generatio
Externí odkaz:
http://arxiv.org/abs/2409.18626
The relationship between a set of design points and the class of hierarchical polynomial models identifiable from the design is investigated. Saturated models are of particular interest. Necessary and sufficient conditions are derived on the set of d
Externí odkaz:
http://arxiv.org/abs/2409.07062
Autor:
Culus, Jean-François, Toulouse, Sophie
Traditionally, inapproximability results for $\mathsf{Max\,k\,CSP\!-\!q}$ have been established using balanced $t$-wise independent distributions, which are related to orthogonal arrays of strength $t$. We contribute to the field by exploring such co
Externí odkaz:
http://arxiv.org/abs/2409.03903
Autor:
Kong, Xiangliang, Tamo, Itzhak
Incidence problems between geometric objects is a key area of focus in the field of discrete geometry. Among them, the study of incidence problems over finite fields have received a considerable amount of attention in recent years. In this paper, by
Externí odkaz:
http://arxiv.org/abs/2408.10977