The weight spectrum of certain affine Grassmann codes (Q670205)

For positive integers $\ell\leq \ell'$ and $m=\ell+\ell'$, let $M_{\ell\times\ell'}$ denote the vector space of all $\ell\times\ell'$ matrices over the finite field $\mathbb{F}_q$. One can view $M_{\ell\times\ell'}$ as an affine subset of the Grassmannian variety $G_{\ell,m}$ by requiring that the Plücker coordinate corresponding to the last $\ell'$ columns is not zero. For the $\ell\times\ell'$ matrix $X=(X_{ij})$ of indeterminates, let $\mathcal{F}^{\mathcal{A}}(\ell,m)$ denote the $\mathbb{F}_q$-span of all $i\times i$ minors of $X$, for $0\leq i\leq \ell$, where the $0\times 0$ minor is the constant polymonomial $1$. The \textit{affine Grassmann code} $C^{\mathcal{A}}(\ell,m)$ is the code obtained by evaluating the elements of $\mathcal{F}^{\mathcal{A}}(\ell,m)$ on the points of $M_{\ell\times\ell'}$. Affine Grassman codes were introduced in [\textit{P. Beelen} et al., IEEE Trans. Inf. Theory 56, No. 7, 3166--3176 (2010; Zbl 1366.94576)] where they calculated their word lenght, dimension and minimum distance. In this article, the authors compute the weight spectrum of the affine Grassmann codes $C^{\mathcal{A}}(2,m)$. In order to do this, first they obtain a relation between the Grassmann code $C(2,m)$, the affine Grassmann code $C^{\mathcal A}(2,m)$, and the Schubert code $C_{\alpha}(2,m)$ associated to the Schubert variety $\Omega_{\alpha}(2,m)$ for $\alpha=(m-2,m)$. Using this relation, they associate a special block-form skew-symmetric matrix to each codeword of $C^{\mathcal{A}}(2,m)$ and from the ranks of some blocks in this matrix they obtain the weight of the codeword. Their main result, Theorem 3, is then proved by counting how many block-form skew-symmetic matrices satisfy certain conditions to determine a codeword.