Finite representations for two small relation algebras

Abstract

In this note, we give two different proofs that relation algebra $52_{65}$ is representable over a finite set. The first is probabilistic, and uses Johnson schemes. The second is an explicit group representation over $(\mathbb{Z}/2\mathbb{Z}{)}^{10}$. We also give a finite representation of $59_{65}$ over $\mathbb{Z}/113\mathbb{Z}$ using a technique due to Comer.