The translation planes of order twenty-five (Q1185900)

The problem of classifying the translation planes of order 25 in three- dimensional projective space \(PG(3,q)\) is equivalent to finding all sets \(M\) of order 25 consisting of \(2\times 2\) matrices over \(GF(5)\) for which any two distinct elements of \(M\) have a non-singular difference. The authors show that, up to isomorphism, there are 21 such sets which they explicitly exhibit. For the corresponding spread sets, five are subregular spread sets, another eight also contain a regulus, and eight do not contain a regulus.