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

TL;DR

This paper introduces $q$-analogues of Fisher's inequality and the oddtown theorem, extending classical combinatorial results to the setting of finite fields with prime power order.

Contribution

It establishes new $q$-analogues of fundamental combinatorial theorems, broadening their applicability in design theory and finite geometry.

Findings

Proves a $q$-analogue of Fisher's inequality.

Presents a $q$-analogue of the oddtown theorem for odd prime powers.

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.