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In this paper we consider the problem of determining the weight spectrum ofq-ary codes C(3,m) associated with Grassmann varieties G(3,m). For m=6 this wasdone by Nogin. We derive a formula for the weight of a codeword of C(3,m), interms of certain varieties associated with alternating trilinear forms on(F_q)^m. The classification of such forms under the action of the generallinear group GL(m,F_q) is the other component that is required to calculate thespectrum of C(3,m). For m=7, we explicitly determine the varieties mentionedabove. The classification problem for alternating 3-forms on (F_q)^7 was solvedby Cohen and Helminck, which we then use to determine the spectrum of C(3,7).

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