Weight Spectrum of Codes Associated With the Grassmannian $G(3,7)$

In this paper, we consider the problem of determining the weight spectrum of <i>q</i> -ary codes <i>C</i>(3,<i>m</i>) associated with Grassmann varieties <i>G</i>(3,<i>m</i>). For <i>m</i>=6, this was done in a paper by Nogin in 1997. We derive a formula for the weight of a codeword of <i>C</i>(3,<i>m</i>), 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 <i>GL</i>(<i>m</i>, F_q) is the other component that is required to calculate the spectrum of <i>C</i>(3,<i>m</i>). For <i>m</i>=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 <i>C</i>(3,7).