Related papers: Almost Designs and Their Links with Balanced Incom…
We study $T$-designs in the nonbinary Johnson scheme. This scheme generalizes both the Johnson and Hamming schemes and admits a bivariate $Q$-polynomial structure. Zhu (2021) provided a combinatorial characterization of $T$-designs in this…
We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an expository treatment of our recently developed…
Combinatorial $t$-designs have nice applications in coding theory, finite geometries and several engineering areas. There are two major methods of constructing $t$-designs. One of them is via group actions of certain permutation groups…
Graphical designs are subsets of vertices of a graph that perfectly average a selected set of eigenvectors of the Graph Laplacian. We show that in highly-structured graphs, graphical designs can coincide with highly structured and…
We propose a procedure of constructing new block designs starting from a given one by looking at the intersections of its blocks with various sets and grouping those sets according to the structure of the intersections. We introduce a…
Block-transitive Steiner $t$-designs form a central part of the study of highly symmetric combinatorial configurations at the interface of several disciplines, including group theory, geometry, combinatorics, coding and information theory,…
Using cyclotomic classes of order twelve for certain finite fields, we construct an infinite family of almost difference sets and normally regular graphs applying the theory of cyclotomy. We show that in each of these fields neither the…
In this paper, we provide some general convergence results for adaptive designs for treatment comparison, both in the absence and presence of covariates. In particular, we demonstrate the almost sure convergence of the treatment allocation…
Modern experimental designs often face the so-called treatment cardinality constraint, which is the constraint on the number of included factors in each treatment. Experiments with such constraints are commonly encountered in engineering…
A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find…
Recently, a construction of group divisible designs (GDDs) derived from the decoding of quadratic residue (QR) codes was given. In this paper, we extend the idea to obtain a new family of GDDs, which is also involved with a well-known…
A linear code with parameters $[n, k, n - k + 1]$ is called maximum distance separable (MDS), and one with parameters $[n, k, n - k]$ is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes…
We study classes of graphs with bounded clique-width that are well-quasi-ordered by the induced subgraph relation, in the presence of labels on the vertices. We prove that, given a finite presentation of a class of graphs, one can decide…
A combinatorial design is a family of sets that are almost disjoint, which is applied in pseudo random number generations and randomness extractions. The parameter, $\rho$, quantifying the overlap between the sets within the family, is…
I give an elementary proof that a quasi-symmetric design without repeated blocks on $v$ points has at most $v\choose2$ blocks, with equality if and only if it is a tight $4$-design.
Motivated by the similarities between the theory of spherical $t$-designs and that of $t$-designs in $Q$-polynomial association schemes, we study two versions of relative $t$-designs, the counterparts of Euclidean $t$-designs for $P$-…
We develop almost ring theory, which is a domain of mathematics somewhere halfway between ring theory and category theory (whence the difficulty of finding appropriate MSC-class numbers). We apply this theory to valuation theory and to…
Hedayat et al. first introduced balanced sampling plans for the exclusion of contiguous units. Wright detailed the results of a preliminary investigation of two-dimensional balanced sampling plans avoiding adjacent units (2-BSAs), and…
The aim of this paper is to present a recursive construction of simple t-designs for arbitrary t. The construction is of purely combinatorial nature and it requires finding solutions for the indices of the ingredient designs that satisfy a…
Computational techniques for the construction of quasi-symmetric block designs are explored and applied to the case with $56$ points. One new $(56,16,18)$ and many new $(56,16,6)$ designs are discovered, and non-existence of $(56,12,9)$ and…