Related papers: Relative t-designs in binary Hamming association s…
We complete the classification of tight $4$-designs in Hamming association schemes $H(n,q)$, i.e., that of tight orthogonal arrays of strength $4$, which had been open since a result by Noda (1979). To do so, we construct an association…
We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed…
Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting…
In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the $t$-adesign, which was coined by Cunsheng Ding in 2015. It is clear that $2$-adesigns are a kind of partially balanced…
Combinatorial $t$-designs have wide applications in coding theory, cryptography, communications and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a $t$-design. In this paper, we first…
In this paper, we examine a class of doubly resolvable combinatorial objects. Let $t, k, \lambda, s$ and $v$ be nonnegative integers, and let $X$ be a set of $v$ symbols. A generalized Howell design, denoted $t$-$GHD_{k}(s,v;\lambda)$, is…
Near maximum distance separable (NMDS) codes, where both the code and its dual are almost maximum distance separable, play pivotal roles in combinatorial design theory and cryptographic applications. Despite progress in fixed dimensions…
A finite subset $Y$ on the unit sphere $S^{n-1} \subseteq \mathbb{R}^n$ is called a spherical design of harmonic index $t$, if the following condition is satisfied: $\sum_{\mathbf{x}\in Y}f(\mathbf{x})=0$ for all real homogeneous harmonic…
We extend the construction of the T-duality symmetry for the 2d compact boson to arbitrary values of the radius by including topological manipulations such as gauging continuous symmetries with flat connections. We show that the entire…
The interplay between coding theory and $t$-designs has attracted a lot of attention for both directions. It is well known that the supports of all codewords with a fixed weight in a code may hold a $t$-design. In this paper, by determining…
We develop the concept of a unitary t-design as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group U(2^n) on n qubits. In particular, sets of unitaries forming…
A Euclidean $t$-design, as introduced by Neumaier and Seidel (1988), is a finite set ${\cal X} \subset \mathbb{R}^n$ with a weight function $w: {\cal X} \rightarrow \mathbb{R}^+$ for which $$\sum_{r \in R} W_r \overline{f}_{S_{r}} =…
Generalized $t$-designs, which form a common generalization of objects such as $t$-designs, resolvable designs and orthogonal arrays, were defined by Cameron [P.J. Cameron, A generalisation of $t$-designs, \emph{Discrete Math.}\ {\bf 309}…
It is known that a Delsarte $t$-design in a $Q$-polynomial association scheme has degree at least $\left \lceil{\frac{t}{2}}\right \rceil $. Following Ionin and Shrikhande who studied combinatorial $(2s-1)$-designs (i.e., Delsarte designs…
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…
In this paper, we provide a criterion for determining whether multiple shells support a $t$-design. We construct as a corollary an infinite series of $2$-designs using power residue codes.
Recall that combinatorial $2s$-designs admit a classical lower bound $b \ge \binom{v}{s}$ on their number of blocks, and that a design meeting this bound is called tight. A long-standing result of Bannai is that there exist only finitely…
We develop a marking system for an analog of Hasse diagrams of intervals $[u,v]$ with $u\leq v$ in a Hermitian symmetric pair $W/W_J$, and use this to create a closed form algorithm for computing relative R-polynomials. The uniform nature…
In this paper we construct two new symmetric designs with parameters 2-(176,50,14) as designs invariant under certain subgroups of the full automorphism group of the Higman design. One is self-dual and has the full automorphism group of…
We generalise the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes. This general framework includes the problem of decomposing hypergraphs with…