Constant-Rank Codes

Abstract

Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random network coding. In this paper, we show that constant-rank codes are closely related to constant-dimension codes and we study the properties of constant-rank codes. We first introduce a relation between vectors in $\mathrm{GF}(q^m{)}^n$ and subspaces of $\mathrm{GF}(q{)}^m$ or $\mathrm{GF}(q{)}^n$, and use it to establish a relation between constant-rank codes and constant-dimension codes. We then derive bounds on the maximum cardinality of constant-rank codes with given rank weight and minimum rank distance. Finally, we investigate the asymptotic behavior of the maximal cardinality of constant-rank codes with given rank weight and minimum rank distance.

5 pages, to appear in Proc. IEEE ISIT 2008