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 a paper by Nogin in 1997. 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^m. The classification of such forms under the action of the general linear group GL(m, F_q) 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 three-forms on F_q⁷ was solved in a study by Cohen and Helminck in 1988, which we then use to determine the spectrum of C(3,7).