Related papers: Steiner t-designs for large t

Let $\Psi(t,k)$ denote the set of pairs $(v,\lambda)$ for which there exists a graphical $t$-$(v,k,\lambda)$ design. Most results on graphical designs have gone to show the finiteness of $\Psi(t,k)$ when $t$ and $k$ satisfy certain…

Interplay between coding theory and combinatorial $t$-designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of $3$-designs have been…

Block designs are combinatorial structures in which each pair of a set of varieties appears together in a fixed number of blocks. Complete graphs are graphs in which every pair of vertices are adjacent. We present some new constructions of…

We clarify the mathematical structure underlying unitary $t$-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any $t$-th order polynomial over the design equals the average over the entire…

Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct…

A weighted $t$-design in $\mathbb{R}^d$ is a finite weighted set that exactly integrates all polynomials of degree at most $t$ with respect to a given probability measure. A fundamental problem is to construct weighted $t$-designs with as…

Let $n \in \mathbb{N}_{\geq 2}$. We prove that for every $k \geq 4$ there exist uniform but non-homogeneous Steiner bundles on $\mathbb{P}^n$ of $k$-type with disconnected splitting type, and we further investigate almost-uniform Steiner…

We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in…

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial $t$-designs have been attracted lots of research interest for decades. The interplay between coding theory and $t$-designs has…

Combinatorial $t$-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and $t$-designs has been…

We present a recursive construction of a (2t + 1)-wise uniform set of permutations on 2n objects using a (2t + 1) - (2n, n, \cdot) combinatorial design, a t-wise uniform set of permutations on n objects and a (2t+1)-wise uniform set of…

We classify bounded t-structures on the category of perfect complexes over a commutative, Noetherian ring of finite Krull dimension, extending a result of Alonso Tarrio, Jeremias Lopez and Saorin which covers the regular case. In…

This paper gives a construction of group divisible designs on the binary extension fields with block sizes 3, 4, 5, 6, and 7, respectively, which is motivated from the decoding of binary quadratic residue codes. A conjecture is proposed for…

For a subset T of nodes of an undirected graph G, a T-Steiner cut is a cut \delta(S) where S intersects both T and the complement of T. The T-Steiner cut dominant} of G is the dominant CUT_+(G,T) of the convex hull of the incidence vectors…

Unitary designs are essential tools in several quantum information protocols. Similarly to other design concepts, unitary designs are mainly used to facilitate averaging over a relevant space, in this case, the unitary group…

In this article, we study $2$-designs with $\gcd(r, \lambda)=1$ admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine all pairs…

Derrick's theorem on the nonexistence of stable time-independent scalar field configurations [G. H. Derrick, J. Math. Phys. 5, 1252 (1964)] is generalized to finite systems of arbitrary dimension. It is shown that the "dilation" argument…

Let $M(n,d)$ be the maximum size of a permutation array on $n$ symbols with pairwise Hamming distance at least $d$. Some permutation arrays can be constructed using blocks of certain type [2] called product blocks in this paper. We study…

One of the very first results about designs over finite fields, by S. Thomas, is the existence of a cyclic 2-$(n,3,7)$ design over $\mathbb{F}_{2}$ for every integer $n$ coprime with 6. Here, by means of difference methods, we reprove and…

In an influential 2008 paper, Baker proposed a number of conjectures relating the divisor theory of algebraic curves with an analogous combinatorial theory on finite graphs. In this note, we examine Baker's Brill--Noether existence…