Weight spectrum of codes associated with the Grassmannian G(3,7)

In this paper we consider the problem of determining the weight spectrum of q-ary codes C(3,m) associated with Grassmann varieties G(3,m). For m = 6 this was done in [1]. We derive a formula for the weight of a codeword of C(3,m), in terms of certain varieties associated with alternating trilinear forms on F q . The classification of such forms under the action of the general linear group GL(m,Fq) is the other component that is required to calculate the spectrum of C(3,m). For m = 7, we explicitly determine the varieties mentioned above. The classification problem for alternating 3-forms on F was solved in [2] which we then use to determine the spectrum of C(3, 7).