A genealogy of the translation planes of order 25

TL;DR

This paper classifies all translation planes of order 25, updating previous classifications with new construction techniques, and identifies two unique planes not fitting existing families, encouraging further research.

Contribution

It provides the first comprehensive classification of order 25 translation planes in light of recent construction methods, identifying two novel planes outside known families.

Findings

Two planes do not belong to any known infinite family.

Updated classification against current construction techniques.

Provides detailed context for the two unique planes.

Abstract

In 1992 Czerwinski and Oakden (The translation planes of order 25, J. Combin. Theory Ser. A, 59:193-217, 1992) provided an exhaustive list of all spreads of $PG(3,5)$ and thus of all translation planes of that order. At that time, the authors provided a partial correlation of these planes to those then-described in the literature, but the intervening years have provided additional construction techniques and classification results. This paper provides an extensive classification of these planes against the currently-known construction techniques, finding two planrs that do not belong to any current infinite family. The author provides additional details for these two planes to help put them in context, as a spur for further research.