On the Similarities Between Generalized Rank and Hamming Weights and Their Applications to Network Coding

Abstract

Rank weights and generalized rank weights have been proved to characterize error and erasure correction, and information leakage in linear network coding, in the same way as Hamming weights and generalized Hamming weights describe classical error and erasure correction, and information leakage in wire-tap channels of type II and code-based secret sharing. Although many similarities between both the cases have been established and proved in the literature, many other known results in the Hamming case, such as bounds or characterizations of weight-preserving maps, have not been translated to the rank case yet, or in some cases have been proved after developing a different machinery. The aim of this paper is to further relate both weights and generalized weights, show that the results and proofs in both cases are usually essentially the same, and see the significance of these similarities in network coding. Some of the new results in the rank case also have new consequences in the Hamming case.