We study the problem of reducing the communication overhead from a noisy
wire-tap channel or storage system where data is encoded as a matrix, when more
columns (or their linear combinations) are available. We present its
applications to reducing communication overheads in universal secure linear
network coding and secure distributed storage with crisscross errors and
erasures and in the presence of a wire-tapper. Our main contribution is a
method to transform coding schemes based on linear rank-metric codes, with
certain properties, to schemes with lower communication overheads. By applying
this method to pairs of Gabidulin codes, we obtain coding schemes with optimal
information rate with respect to their security and rank error correction
capability, and with universally optimal communication overheads, when n≤m, being n and m the number of columns and number of rows,
respectively. Moreover, our method can be applied to other families of maximum
rank distance codes when n>m. The downside of the method is generally
expanding the packet length, but some practical instances come at no cost.Comment: 21 pages, LaTeX; parts of this paper have been accepted for
presentation at the IEEE International Symposium on Information Theory,
Aachen, Germany, June 201