On the largest subsets avoiding the diameter of $(0,\pm 1)$-vectors

TL;DR

This paper classifies the largest subsets of specific vector sets with restricted entries that have smaller diameter than the original, solving an open problem related to 4-distance sets and Johnson schemes.

Contribution

It provides a classification of the largest subsets with smaller diameter for certain parameters, including an open problem in the theory of distance sets and Johnson schemes.

Findings

Classified largest subsets for m=1, l=2, any k

Identified largest 4-distance sets containing Johnson scheme J(9,4)

Solved an open problem from Bannai, Sato, and Shigezumi (2012)

Abstract

Let $L_{mkl}\subset \mathbb{R}^{m+k+l}$ be the set of vectors which have $m$ of entries $-1$, $k$ of entries $0$, and $l$ of entries $1$. In this paper, we investigate the largest subset of $L_{mkl}$ whose diameter is smaller than that of $L_{mkl}$. The largest subsets for $m=1$, $l=2$, and any $k$ will be classified. From this result, we can classify the largest $4$-distance sets containing the Euclidean representation of the Johnson scheme $J(9,4)$. This was an open problem in Bannai, Sato, and Shigezumi (2012).