An Assmus–Mattson theorem for rank metric codes

Eimear Byrne, Alberto Ravagnani

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)

Abstract

A t-(n, d, λ) design over Fq, or a subspace design, is a collection of d-dimensional subspaces of Fn q , called blocks, with the property that every t-dimensional subspace of Fn q is contained in the same number λ of blocks. A collection of n × m matrices over Fq is said to hold a t-design over Fq if the set of column spaces of its elements forms the blocks of a subspace design. We use notions of puncturing and shortening of rank metric codes and the rank metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold a t-design over Fq. We show that for Fqm-linear vector rank metric codes, the property of a code being maximum rank distance (MRD) is equivalent to its minimal weight codewords holding trivial subspace designs, and show that this characterization does not hold for Fq-linear matrix MRD codes that are not linear over Fqm. Finally, using arguments based on covering radius and external distance, we establish various existence results that apply to both the rank and the Hamming metric.

Original languageEnglish
Pages (from-to)1242-1260
Number of pages19
JournalSIAM Journal on Discrete Mathematics
Volume33
Issue number3
DOIs
Publication statusPublished - 2019
Externally publishedYes

Keywords

  • Assmus-Mattson theorem
  • Design over Fq
  • MacWilliams identities
  • Rank metric code
  • Subspace design
  • Weight distribution

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