A note on spanoid rank

TL;DR

This paper constructs a specific spanoid with a rank significantly larger than its f-rank, answering an open question and demonstrating a notable separation between these two measures.

Contribution

It provides a construction of a spanoid on n elements with a rank at least n^c times its f-rank, where c is approximately 0.113283, addressing a question posed by Dvir-Gopi-Wigderson.

Findings

Constructed a spanoid with rank ≥ n^c * f-rank

Established a separation between rank and f-rank measures

Answered an open question in the study of spanoids

Abstract

We construct a spanoid $\mathcal{S}$ on $n$ elements with $\textsf{rank}(\mathcal{S}) \ge n^c \textsf{f-rank}(\mathcal{S})$ where $c = \log_5 3 - \log_5 2.5 \approx 0.113283$. This answers a question of Dvir-Gopi-Wigderson [DGW18].