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Decomposable subspaces, linear sections of Grassmann varieties, and higher weights of Grassmann codes. (English) Zbl 1155.14019
Given integers \(\ell \leq m\) and an \(m\)-dimensional vector space \(V\) over a finite field \(F=GF(q)\), let \(G_{\ell,m}\) denote the Grassmannian of all \(\ell\)-dimensional subspaces of \(V\). This can (and will be) identified with a subspace of \(\mathbb{P}^1(\wedge^\ell V)\). Define the Grassmannian code \(C=C(\ell,m)\) to be the \([n,k]_q\) code corresponding to the projective system associated to the Plücker embedding \(G_{\ell,m}\mapsto \mathbb{P}^{k-1}(F)\), where \(n = \left[ \begin{smallmatrix} m\\ \ell \end{smallmatrix} \right]_q\) is the Gauss binomial coefficient, \(k = \left( \begin{smallmatrix} m\\ \ell \end{smallmatrix} \right)\) is the usual binomial coefficient.
Define the \(r\)-th weight of \(C\) by \(d_r(C) = n-\max_{L} |L\cap G_{\ell,m}|\), where \(n\) s as above and \(L\) runs over all linear subvarieties of \(\mathbb{P}^1(\wedge^\ell V)\). Though much research has been done on these higher weights, the complete determination of them is open. The authors begin their paper with an excellent survey of the literature and then proves new results in the case \(\ell = 2\) using new techniques. A well-written paper.

MSC:
14G15 Finite ground fields in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
14G50 Applications to coding theory and cryptography of arithmetic geometry
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