The authors produce an infinite class of combinatorial objects which they call near biplanes. A near biplane is defined as follows: it is a square 1-design with the property that its \(V\times V\) incidence matrix satisfies (i) the inner product of any two distinct rows is 0, 2 or 4; (ii) for any fixed row the sum of the inner products with the remaining rows is \(2(v-1).\) The main theorem is reduced to the following statement: (a) For \(q=4\), either choice of orbit gives rise to a biplane with parameters \(v=11\), \(k=5\), \(\lambda =2;\) (b) For \(q=8\), there is a unique choice of orbit (from the four possible) which gives rise to a biplane with parameters \(v=37\), \(k=9\), \(\lambda =2;\) (c) For \(q>8\), we never obtain a biplane (for any choice of orbit).