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 13 of twentysix vectors in the 13-dimensional vector space V = GF(5)13, 13 , with the property that there exist precisely thirteen 6-subsets of 13 whose elements sum up to zero in V , which can also be characterized as the intersections of 13 with thirteen linear hyperplanes of V .