Tight relative $t$-designs on two shells in hypercubes, and Hahn and Hermite polynomials

TL;DR

This paper investigates tight relative $t$-designs on two shells in hypercubes, revealing their algebraic structure and linking polynomial zeros to design rarity, using Terwilliger algebra and orthogonal polynomials.

Contribution

It establishes that such designs induce coherent configurations and connects polynomial zeros to design existence, providing new insights into their rarity for large $t$.

Findings

Designs induce coherent configurations with two fibers.

Polynomial zeros are integral and simple under certain conditions.

Tight relative $t$-designs are rare for large $t$.

Abstract

Relative $t$-designs in the $n$-dimensional hypercube $\mathcal{Q}_n$ are equivalent to weighted regular $t$-wise balanced designs, which generalize combinatorial $t$-$(n,k,\lambda)$ designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean $t$-designs on two concentric spheres, in this paper we discuss tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells. We show under a mild condition that such a relative $t$-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells are rare for large $t$.