Isodiametric inequality for vector spaces

TL;DR

This work extends Kleitman’s isodiametric inequality from Hamming space to the subspace lattice over $\mathbb{F}_q^n$ by establishing a $q$-analog on the $n$-dimensional $q$-Hamming graph. The authors prove that, for large $n$ relative to the diameter $d$, any family $\mathcal{F}$ with $\mathrm{diam}(\mathcal{F})\le d$ satisfies $|\mathcal{F}|\le \sum_{i=0}^{t} \binom{n}{i}_q$ if $d=2t$, and $|\mathcal{F}|\le \sum_{i=0}^{t} \binom{n}{i}_q+\binom{n-1}{t}_q$ if $d=2t+1$, with optimal extremal families described. The proof combines an isometry from orthogonal complementation, perfect-matchings arguments in $q$-Hamming graphs, and Erdős–Ko–Rado–type bounds for intersecting subspace families, followed by a careful division into cases $n=d+1$, $d=2,3$, and $d\ge 4$ to handle all parameter regimes. The results unify discrete isodiametric phenomena in the subspace setting and suggest broader applicability of these techniques to other $q$-analog combinatorial geometries. This contributes to our understanding of extremal set systems in vector spaces and offers precise extremal structures for large $n$.

Abstract

A theorem of Kleitman states that a collection of binary vectors with diameter d has cardinality at most that of a Hamming ball of radius d/2. In this paper, we give a q-analog of it.

Theorems & Definitions (14)

• Theorem 1.1: Kleitman, MR200179
• Theorem 1.2
• Lemma 2.1: MR3497070
• Proposition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• Theorem 2.6: MR867648