An algebraic representation of Steiner triple systems of order 13

In this paper we construct an incidence structure isomorphic to a Steiner triple system of order 13 by defining a set B of twentysix vectors in the 13-dimensional vector space V = GF(5)^13, with the property that there exist precisely thirteen 6-subsets of B whose elements sum up to zero in V, which can also be characterized as the intersections of B with thirteen linear hyperplanes of V.