Error-correcting codes in projective spaces via rank-metric codes and ferrers diagrams

Tuvi Etzion, Natalia Silberstein

Research output: Contribution to journalArticlepeer-review

141 Scopus citations


Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced row echelon form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper, we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant-weight code. Each codeword defines a skeleton of a basis for a subspace in reduced row echelon form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rank-metric code is lifted to a constant-dimension code. The union of these codes is our final constant-dimension code. In particular, the codes constructed recently by Koetter and Kschischang are a subset of our codes. The rank-metric codes used for this construction form a new class of rank-metric codes. We present a decoding algorithm to the constructed codes in the projective space. The efficiency of the decoding depends on the efficiency of the decoding for the constant-weight codes and the rank-metric codes. Finally, we use puncturing on our final constant-dimension codes to obtain large codes in the projective space which are not constant-dimension.

Original languageEnglish
Pages (from-to)2909-2919
Number of pages11
JournalIEEE Transactions on Information Theory
Issue number7
StatePublished - 15 Jul 2009
Externally publishedYes


  • Constant-dimension codes
  • Constant-weight codes
  • Ferrers diagram
  • Identifying vector
  • Network coding
  • Projective space codes
  • Puncturing
  • Rank-metric codes
  • Reduced row echelon form

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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