Constant-rank codes and their connection to constant-dimension codes

  • Authors:

  • Maximilien Gadouleau;Zhiyuan Yan

  • Affiliations:

  • School of Electronic Engineering and Computer Science, Queen Mary, University of London, London, UK and Department of Electrical and Computer Engineering, Lehigh University, Bethlehem, PA and CReS ...;Department of Electrical and Computer Engineering, Lehigh University, Bethlehem, PA

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2010

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Abstract

Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random linear network coding. What the maximal cardinality of any constant-dimension code with finite dimension and minimum distance is and how to construct the optimal constant-dimension code (or codes) that achieves the maximal cardinality both remain open research problems. In this paper, we introduce a new approach to solving these two problems. We first establish a connection between constant-rank codes and constant-dimension codes. Via this connection, we show that optimal constant-dimension codes correspond to optimal constant-rank codes over matrices with sufficiently many rows. As such, the two aforementioned problems are equivalent to determining the maximum cardinality of constant-rank codes and to constructing optimal constant-rank codes, respectively. To this end, we then derive bounds on the maximum cardinality of a constant-rank code with a given minimum rank distance, propose explicit constructions of optimal or asymptotically optimal constant-rank codes, and establish asymptotic bounds on the maximum rate of a constant-rank code.