External Difference Families Arising from Two or Three Cyclotomic Classes

We study external difference families arising from cyclotomic classes in finite fields from the viewpoint of a fixed number of blocks. For families consisting of even-indexed cyclotomic classes, the EDF condition can be expressed in terms of relations among cyclotomic numbers. We first study the two-block case and recover a classical characterization in terms of quadratic forms. Our main result shows that, for a prime $p=12k+1$, the family $\{C_0^6,C_2^6,C_4^6\}$ forms an EDF in ${\mathbb{F}}_p$ if and only if $k$ is a square. The proofs combine symmetry relations of cyclotomic numbers with their explicit evaluations.