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
_q
⁷
was solved in a study by Cohen and Helminck in 1988, which we then use to determine the spectrum of
C
(3,7).