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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 Fqm. 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 three-forms on Fq7 was solved in a study by Cohen and Helminck in 1988, which we then use to determine the spectrum of C(3,7).