Equality in the linear algebra bound

TL;DR

This paper investigates conditions under which the linear algebra bound is tight, revealing insights into extremal configurations and applications to set families and Euclidean 2-distance sets.

Contribution

It provides new results on when the linear algebra bound is achieved, especially in the context of set intersections, Hamming distances, and Euclidean space configurations.

Findings

Identifies conditions for equality in the linear algebra bound.

Provides insights into extremal set families with intersection or distance constraints.

Applies findings to 2-distance sets in Euclidean spaces.

Abstract

We study some examples when there is actually an equality in the linear algebra bound. When the vectors considered span in fact the entire space. We would like to point out that in some cases this provides some interesting extra information about the extremal configuration. We obtain results on set families satisfying conditions on pairwise intersections, or Hamming distances. Also, we have an application to 2-distance sets in Euclidean spaces.