On the existence of tight relative 2-designs on binary Hamming association schemes

TL;DR

This paper investigates the structure and existence of tight relative 2-designs on two shells within binary Hamming association schemes, establishing their connection to coherent configurations and identifying cases with non-constant weights.

Contribution

It proves that tight relative 2-designs on two shells in Hamming schemes form coherent configurations and classifies their parameters for n up to 30, including cases with non-constant weights.

Findings

Tight relative 2-designs on two shells are coherent configurations.

Classification of parameters for n ≤ 30.

Existence of designs with non-constant weights for n ≡ 6 mod 8.

Abstract

It is known that there is a close analogy between "Euclidean t-designs vs. spherical t-designs" and "Relative t-designs in binary Hamming association schemes vs. combinatorial t-designs". In this paper, we want to prove how much we can develop a similar theory in the latter situation, imitating the theory in the former one. We first prove that the weight function is constant on each shell for tight relative t-designs on p shells on a wide class of Q-polynomial association schemes, including Hamming association schemes. In the theory of Euclidean t-designs on 2 concentric spheres (shells), it is known that the structure of coherent configurations is naturally attached. However, it seems difficult to prove this claim in a general context. In the case of tight 2-designs in combinatorial 2-designs, there are great many tight 2-designs, i.e., symmetric 2-designs, while there are very few tight 2e-designs for e no less than 2. So, as a starting point, we concentrate our study to the existence problem of tight relative 2-designs, in particular on 2 shells, in binary Hamming association schemes H(n,2). We prove that every tight relative 2-designs on 2 shells in H(n,2) has the structure of coherent configuration. We determined all the possible parameters of coherent configurations attached to such tight relative 2-designs for n at most 30. Moreover for each of them we determined whether there exists such a tight relative 2-design or not, either by constructing them from symmetric 2-designs or Hadamard matrices, or theoretically showing the non-existence. In particular, we show that for n congruent to 6 (mod 8), there exist such tight relative 2-designs whose weight functions are not constant. These are the first examples of those with non-constant weight.