Linear codes associated to determinantal varieties

Peter Beelen, Sudhir R. Ghorpade, Sartaj Ul Hasan

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

We consider a class of linear codes associated to projective algebraic varieties defined by the vanishing of minors of a fixed size of a generic matrix. It is seen that the resulting code has only a small number of distinct weights. The case of varieties defined by the vanishing of 2×2 minors is considered in some detail. Here we obtain the complete weight distribution. Moreover, several generalized Hamming weights are determined explicitly and it is shown that the first few of them coincide with the distinct nonzero weights. One of the tools used is to determine the maximum possible number of matrices of rank 1 in a linear space of matrices of a given dimension over a finite field. In particular, we determine the structure and the maximum possible dimension of linear spaces of matrices in which every nonzero matrix has rank 1.
Original languageEnglish
JournalDiscrete Mathematics
Volume338
Issue number8
Pages (from-to)1493-1500
ISSN0012-365X
DOIs
Publication statusPublished - 2015

Keywords

  • Linear codes
  • Determinantal varieties
  • Generalized Hamming weight
  • Weight distribution

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