On minimal subspace Zp-null designs

TL;DR

This paper investigates minimal non-zero counts in $Z_p$-null designs over finite fields, establishing exact values for binary cases and bounds for larger fields, advancing combinatorial design theory.

Contribution

It determines the minimal non-zero size of $Z_p$-null designs for binary fields and provides bounds for larger fields, extending understanding of combinatorial designs over finite fields.

Findings

Exact minimal non-zero count for $q=p=2$ is $2^{t+1}$.

Provides lower and upper bounds for the minimal size when $q>2$.

Advances the theory of $Z_p$-null designs in finite vector spaces.

Abstract

Let $q$ be a power of a prime $p$, and let $V$ be an $n$-dimensional space over the field GF$(q)$. A $Z_p$-valued function $C$ on the set of $k$-dimensional subspaces of $V$ is called a $k$-uniform $Z_p$-null design of strength $t$ if for every $t$-dimensional subspace $y$ of $V$ the sum of $C$ over the $k$-dimensional superspaces of $y$ equals $0$. For $q=p=2$ and $0\le t<k<n$, we prove that the minimum number of non-zeros of a non-void $k$-uniform $Z_p$-null design of strength $t$ equals $2^{t+1}$. For $q>2$, we give lower and upper bounds for that number.