zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv
[1] Barnabei, M.; Brini, A.; Rota, G.-C., On the exterior calculus of invariant theory, J. algebra, 96, 120-160, (1985) · Zbl 0585.15005
[2] Bourbaki, N., Elements of mathematics: algebra I, (1974), Hermann Paris, Chapters 1-3
[3] Cummings, L.J., Decomposable symmetric tensors, Pacific J. math., 35, 65-77, (1970) · Zbl 0197.30603
[4] Ghorpade, S.R.; Lachaud, G., Higher weights of Grassmann codes, (), 122-131 · Zbl 1021.94026
[5] Ghorpade, S.R.; Lachaud, G., Hyperplane sections of Grassmannians and the number of MDS linear codes, Finite fields appl., 7, 468-506, (2001) · Zbl 1007.94024
[6] Ghorpade, S.R.; Tsfasman, M.A., Schubert varieties, linear codes and enumerative combinatorics, Finite fields appl., 11, 684-699, (2005) · Zbl 1147.94014
[7] Hansen, J.P.; Johnsen, T.; Ranestad, K., Schubert unions in Grassmann varieties, Finite fields appl., 13, 738-750, (2005), longer version in · Zbl 1136.14036
[8] J.P. Hansen, T. Johnsen, K. Ranestad, Grassmann codes and Schubert unions, in: Arithmetic, Geometry and Coding Theory, AGCT-2005, Luminy, in: Séminaires et Congrès, Soc. Math. France, Paris, in press · Zbl 1218.14042
[9] Hirschfeld, J.W.P.; Thas, J.A., General Galois geometries, (1991), Oxford Univ. Press Oxford · Zbl 0789.51001
[10] Helleseth, T.; Kløve, T.; Levenshtein, V.I.; Ytrehus, Ø., Bounds on the minimum support weights, IEEE trans. inform. theory, 41, 432-440, (1995) · Zbl 0842.94021
[11] Lim, M.H., A note on maximal decomposable subspaces of symmetric spaces, Bull. London math. soc., 7, 289-293, (1975) · Zbl 0327.15033
[12] Marcus, M., Finite dimensional multilinear algebra, part II, (1975), Marcel Dekker New York · Zbl 0339.15003
[13] MacWilliams, F.J.; Sloane, N.J.A., The theory of error correcting codes, (1977), North-Holland Amsterdam · Zbl 0369.94008
[14] Nogin, D.Yu., Codes associated to Grassmannians, (), 145-154 · Zbl 0865.94032
[15] Parthasarathy, K.R., On the maximal dimension of a completely entangled subspace for finite level quantum systems, Proc. Indian acad. sci. math. sci., 114, 1-10, (2004) · Zbl 1080.46014
[16] Tanao, H., On \((n - 1)\)-dimensional projective spaces contained in the Grassmann variety \(\operatorname{Gr}(n, 1)\), J. math. Kyoto univ., 14, 415-460, (1974)
[17] Tsfasman, M.A.; Vlăduţ, S.G., Geometric approach to higher weights, IEEE trans. inform. theory, 41, 1564-1588, (1995) · Zbl 0853.94023
[18] Wei, V.K., Generalized Hamming weights for linear codes, IEEE trans. inform. theory, 37, 1412-1418, (1991) · Zbl 0735.94008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.