$q$-analogues of Fisher's inequality and oddtown theorem

TL;DR

The paper develops $q$-analogues of Fisher's inequality and the oddtown theorem in the setting of subspaces over a finite field of order $q$. It introduces an incidence-vector method based on the $[n]_q$ one-dimensional subspaces to relate intersection dimensions to inner products $<f_A,f_B>=[dim(A\cap B)]_q$. It proves a $q$-analogue of Fisher's inequality, $|\mathcal{F}|\le [n]_q$, and a $q$-analogue of oddtown for odd prime powers with the same bound, plus a reverse oddtown analogue with parity-dependent bounds; tightness is shown by simple families such as all 1-dimensional subspaces, and constructions are discussed along with conjectures for even $n$ and even $q$. These results advance extremal combinatorics in finite geometry and lay groundwork for further $q$-analogues.

Abstract

A classical result in design theory, known as Fisher's inequality, states that if every pair of clubs in a town shares the same number of members, then the number of clubs cannot exceed the number of inhabitants in the town. In this short note, we establish a $q$-analogue of Fisher's inequality. Additionally, we present a $q$-analogue of the oddtown theorem for the case when $q$ is an odd prime power.

Theorems & Definitions (18)

• Theorem 1.1: Fisher
• Theorem 1.2
• Theorem 1.3: Oddtown theorem
• Theorem 1.4
• Theorem 1.5: Reverse oddtown theorem
• Theorem 1.6
• Remark 1.7
• Conjecture 1.8
• Lemma 2.1
• proof