The odd and even sets of $$\textrm{PG}(2,8)$$, up to equivalence

An even (resp. odd) set in a projective plane is a set of points such that every line of the plane intersects that set in an even (resp. odd) number of points. We obtained, by computer, a full classification up to equivalence of the even and odd sets of the projective plane of order 8. The generation algorithm is non-standard in that it does not incrementally add points to previously generated sets but instead starts from a family of ‘irreducible’ sets and takes subsequent symmetric differences with lines. We give geometric descriptions of the resulting sets with the largest automorphism groups, sets that can be constructed from bundles of hyperovals, linear sets, sets related to subplanes, and others. We also provide the weight enumerator of the linear code whose elements are the characteristic vectors of all even sets. We include a computer-free proof of the classification of odd and even sets up to size 13.