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

Abstract

Relative t-designs in the n-dimensional hypercube Qn are equivalent to weighted regular t-wise balanced designs, which generalize combinatorial t-(n, k, λ) 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 Qn 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 Qn supported on two shells are rare for large t.

$Q$-polynomial coherent configurations

“…-design in J(n, i) for i ∈ {ℓ, m} and the degree s i,j between Y i and Y j is at most e. It was shown in [5,Theorem 5.3…”

“…We will claim that the coherent configuration is Q-polynomial based on [25]. Furthermore it was shown in [5] that tight relative 2e-designs on two shells in the binary Hamming scheme H(n, 2) yield a Q-polynomial coherent configurations. Motivated by this work, we generalize Theorem 5.5 to designs in fibers of a Q-polynomial coherent configuration.…”

“…It was shown in [5,Theorem 5.3] that a tight relative t-design in H(n, 2) = (X, {R i } n i=0 ) on two shells yields a coherent configuration. We include the result and claim that the resulting coherent configuration is Q-polynomial.…”

“…In Section 5, several examples of Q-polynomial coherent configurations are provided from Q-antipodal Q-polynomial association schemes, complete orthogonal array of strength 4, tight spherical t-designs for t = 4, 5, 7, and maximal mutually unbiased bases. Section 6 is taken from [25] and [5]. It is known that the Terwilliger algebra of the binary Hamming schemes is a coherent configuration.…”