Decomposable subspaces,linear sections of Grassmann varieties,and higher weights of Grassmann codes |
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Authors: | Sudhir R Ghorpade Arunkumar R Patil Harish K Pillai |
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Institution: | 1. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India;2. Shri Guru Gobind Shinghji Institute of Engineering & Technology, Vishnupuri, Nanded 431 606, India;3. Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India |
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Abstract: | We consider the question of determining the maximum number of points on sections of Grassmannians over finite fields by linear subvarieties of the Plücker projective space of a fixed codimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties. We recover most of the known results as well as prove some new results. A basic tool used is a characterization of decomposable subspaces of exterior powers, that is, subspaces in which every nonzero element is decomposable. Also, we use a generalization of the Griesmer–Wei bound that is proved here for arbitrary linear codes. |
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