A Four-Dimensional Kerdock Set over GF(3)

Abstract If m is an even integer and K = GF(q) is a field of characteristic 2, then there exists a set of qm−1 alternating bilinear forms of degree m over K such that the difference of any two of the forms is nonsingular. Do such sets exist over fields of odd characteristic? This note constructs such a set in the smallest nontrivial case, namely, m = 4, q = 3.