Two-Distance Sets over Finite Fields

Abstract

We study two-distance sets in standard $d$-dimensional quadratic spaces over finite fields. In characteristic $3$, we construct sets attaining the full Larman--Rogers--Seidel bound, showing that Blokhuis' Euclidean bound $\left(\genfrac{}{}{0ex}{}{d+2}{2}\right)$ need not hold over finite fields. We prove a rank-sensitive replacement which precisely measures the failure of positive definiteness. We also show that the Blokhuis bound $\left(\genfrac{}{}{0ex}{}{d+2}{2}\right)$ is attained over a suitable finite field exactly for each $d\ne 6$; in the exceptional dimension $d=6$, the exact maximum is $27$.

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