The structure of dual Grassmann codes

Peter Beelen, Fernando Pinero

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Abstract

In this article we study the duals of Grassmann codes, certain codes coming from the Grassmannian variety. Exploiting their structure, we are able to count and classify all their minimum weight codewords. In this classification the lines lying on the Grassmannian variety play a central role. Related codes, namely the affine Grassmann codes, were introduced more recently in Beelen et al. (IEEE Trans Inf Theory 56(7):3166–3176, 2010), while their duals were introduced and studied in Beelen et al. (IEEE Trans Inf Theory 58(6):3843–3855, 2010). In this paper we also classify and count the minimum weight codewords of the dual affine Grassmann codes. Combining the above classification results, we are able to show that the dual of a Grassmann code is generated by its minimum weight codewords. We use these properties to establish that the increase of value of successive generalized Hamming weights of a dual Grassmann code is 1 or 2.
Original languageEnglish
JournalDesigns, Codes and Cryptography
Volume79
Issue number3
Pages (from-to)451-470
ISSN0925-1022
DOIs
Publication statusPublished - 2016

Keywords

  • Dual Grassmann code
  • Hamming weights
  • Tanner code

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