Semidefinite programming bounds for few-distance sets in the Hamming and Johnson spaces

TL;DR

This paper develops semidefinite programming methods to determine the maximum size of sets with few distances in Hamming and Johnson spaces, extending previous work and providing exact bounds for certain cases.

Contribution

It introduces new semidefinite programming formulations that extend prior research, enabling exact determination of maximum sizes for two- and three-distance sets.

Findings

Exact bounds for maximum sizes of two-distance sets in specific parameters

Exact bounds for maximum sizes of three-distance sets in specific parameters

Extension of previous semidefinite programming approaches

Abstract

We study the maximum cardinality problem of a set of few distances in the Hamming and Johnson spaces. We formulate semidefinite programs for this problem and extend the 2011 works by Barg-Musin and Musin-Nozaki. As our main result, we find new parameters for which the maximum size of two- and three-distance sets is known exactly.