Weight Spectrum of Codes Associated With the Grassmannian $G(3,7)$
Krishna V. Kaipa, Harish K. Pillai · IEEE Transactions on Information Theory · 2012
In this paper, we consider the problem of determining the weight spectrum ofq-ary codesC(3,m) associated with Grassmann varietiesG(3,m). Form=6, this was done in a paper by Nogin in 1997. We derive a formula for the weight of a codeword ofC(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 groupGL(m, Fq) is the other component that is required to calculate the spectrum ofC(3,m). Form=7, we explicitly determine the varieties mentioned above. The classification problem for alternating three-forms on Fq7was solved in a study by Cohen and Helminck in 1988, which we then use to determine the spectrum ofC(3,7).