Maximal sets of a given diameter in Hamming cubes

TL;DR

This paper characterizes the structure and size bounds of maximal sets with a fixed diameter in Hamming cubes, revealing their connection to Hamming balls and establishing tight bounds.

Contribution

It proves that d-maximal sets are either small or contain a Hamming ball, and shows the finiteness of essentially different d-maximal sets.

Findings

d-maximal sets are either of size at most (n+o(n))^d or contain a Hamming ball

The bound (n+o(n))^d is asymptotically tight

The number of essentially different d-maximal sets is finite

Abstract

A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or contains a non-trivial Hamming ball. The bound of $(n+o(n))^d$ is asymptotically tight. Additionally, we give a non-trivial lower bound on the size of any $d$-maximal set and show that the number of essentially different $d$-maximal sets is finite.