The Width of Hamming Balls

TL;DR

This paper investigates the width of Hamming ball posets, extending known results from small radii to larger ones, and provides a corrected proof inspired by Harper's work.

Contribution

It generalizes the understanding of the width of Hamming ball posets beyond small radii, correcting previous proofs and extending Sperner-type results.

Findings

Hamming ball posets have width equal to their largest layer.

Extension of Sperner's theorem to Hamming ball posets.

Provides a corrected proof inspired by Harper.

Abstract

The width of a poset is the size of its largest antichain. Sperner's theorem states that $(2^{[n]},\subset)$ is a poset whose width equals the size of its largest layer. We show that Hamming ball posets also have this property. This extends earlier work that proves this in the case of small radii. Our proof is inspired by (and corrects) a result of Harper.