Rank of incidence matrices over integers modulo a prime power

TL;DR

This paper establishes an improved upper bound on the rank over finite fields of incidence matrices in modular integer spaces, specifically for points and hyperplanes in $(Z/p^k Z)^n$, enhancing previous bounds for large $k$.

Contribution

It provides a tighter upper bound on the $F_p$-rank of incidence matrices over $(Z/p^k Z)^n$, advancing understanding in combinatorial matrix theory.

Findings

Derived a new upper bound on the matrix rank for large $k$

Improved upon previous bounds by Laba and Trainer

Applicable to incidence matrices of points and hyperplanes in modular integer spaces

Abstract

In this note we prove an upper bound on the $\mathbb F_p$-rank of the incidence matrix of points and hyperplanes in $(\mathbb Z/p^k \mathbb Z)^n$, improving a recent bound of Laba and Trainer when $k$ is large.