We study the problem of reducing the communication overhead from a noisywire-tap channel or storage system where data is encoded as a matrix, when morecolumns (or their linear combinations) are available. We present itsapplications to reducing communication overheads in universal secure linearnetwork coding and secure distributed storage with crisscross errors anderasures and in the presence of a wire-tapper. Our main contribution is amethod to transform coding schemes based on linear rank-metric codes, withcertain properties, to schemes with lower communication overheads. By applyingthis method to pairs of Gabidulin codes, we obtain coding schemes with optimalinformation rate with respect to their security and rank error correctioncapability, and with universally optimal communication overheads, when $ n \leqm $, being $ n $ and $ m $ the number of columns and number of rows,respectively. Moreover, our method can be applied to other families of maximumrank distance codes when $ n > m $. The downside of the method is generallyexpanding the packet length, but some practical instances come at no cost.
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机译：我们研究了在有更多列（或它们的线性组合）可用的情况下减少从嘈杂的窃听通道或存储系统（将数据编码为矩阵）的通信开销的问题。我们提出了其应用程序，以减少通用安全线性网络编码和带有交叉错误和擦除的安全分布式存储中以及存在窃听器的情况下的通信开销。我们的主要贡献是将具有确定属性的基于线性秩度量代码的编码方案转换为具有较低通信开销的方案的方法。通过将此方法应用于成对的Gabidulin码，我们获得了具有相对于其安全性和秩错误纠正能力的最佳信息率，并且具有普遍最优的通信开销的编码方案，当$ n \ leqm $，分别为$ n $和$ m $列数和行数分别。此外，当$ n> m $时，我们的方法可以应用于其他最大秩距离代码族。该方法的缺点通常是扩展数据包长度，但是一些实际情况是免费的。
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