Projective Space Codes for the Injection Metric

Abstract

In the context of error control in random linear network coding, it is useful to construct codes that comprise well-separated collections of subspaces of a vector space over a finite field. This thesis concerns the construction of non-constant-dimension projective space codes for adversarial error-correction in random linear network coding. The metric used is the so-called injection distance introduced by Silva and Kschischang, which perfectly reflects the adversarial nature of the channel. A Gilbert-Varshamov-type bound for such codes is derived and its asymptotic behaviour is analysed. It is shown that in the limit as the ambient space dimension approaches infinity, the Gilbert-Varshamov bound on the size of non-constant-dimension codes behaves similar to the Gilbert-Varshamov bound on the size of constant-dimension codes contained within the largest Grassmannians in the projective space. Using the code-construction framework of Etzion and Silberstein, new non-constant-dimension codes are constructed; these codes contain more codewords than comparable codes designed for the subspace metric. To our knowledge this work is the first to address the construction of non-constant-dimension codes designed for the injection metric.