On subcodes of generalized second order Reed-Muller codes

We study sets of quadratic forms in m variables over GF (q) such that the difference of any two quadratic forms in the set has rank $m$. It is easily seen that the cardinality of such a set is at most q^m, and we shall give four constructions of such sets which have cardinality exactly q^m (where q is an odd prime power). The first construction works only in the case when m is odd, and uses a BCH code. The second is an ad hoc construction in the case when m = 2 The third construction is due to H. A. Wilbrink, and generalizes the second (though not in an obvious way). The fourth uses symmetric representations of finite fields and is due to G. Serousi and A. Lempel. These sets correspond to subcodes of the generalized second order Reed-Muller code, and we give the weight distribution of these codes.