Relative t-designs in binary Hamming association scheme H(n,2)

TL;DR

This paper studies relative t-designs in binary Hamming schemes, focusing on Fisher inequalities, existence of tight designs, and their connections to classical combinatorial designs, providing new families and open problems.

Contribution

It introduces new results on the structure and parameters of tight relative t-designs, especially on two shells, and links them to classical design theory.

Findings

Proved that constant weight functions imply subsets are (t-1)-designs.

Discovered new families of tight relative t-designs for odd t.

Provided feasible parameter lists for t=3 and t=4 designs up to certain n.

Abstract

A relative t-design in the binary Hamming association schemes H(n,2) is equivalent to a weighted regular t-wise balanced design, i.e., certain combinatorial t-design which allow different sizes of blocks and a weight function on blocks. In this paper, we study relative t-designs in H(n,2), putting emphasis on Fisher type inequalities and the existence of tight relative t-designs. We mostly consider relative t-designs on two shells. We prove that if the weight function is constant on each shell of a relative t-design on two shells then the subset in each shell must be a combinatorial (t-1)-design. This is a generalization of the result of Kageyama who proved this under the stronger assumption that the weight function is constant on the whole block set. Using this, we define tight relative t-designs for odd t, and a strong restriction on the possible parameters of tight relative t-designs in H(n,2). We obtained a new family of such tight relative t-designs, which were unnoticed before. We will give a list of feasible parameters of such relative 3-designs with n up to 100, and then we discuss the existence and/or the non-existence of such tight relative 3-designs. We also discuss feasible parameters of tight relative 4-designs on two shells in H(n,2) with n up 50. In this study we come up with the connection on the topics of classical design theory, such as symmetric 2-designs (in particular 2-(4u-1,2u-1,u-1) Hadamard designs) and Driessen's result on the non-existence of certain 3-designs. We believe the Problem 1 and Problem 2 presented in Section 5.2 open a new way to study relative t-designs in H(n,2). We conclude our paper listing several open problems.