On the similarities between generalized rank and Hamming weights and their applications to network coding

TL;DR

This paper explores the deep connections between rank weights and Hamming weights, demonstrating that many results and proofs are fundamentally similar, which enhances understanding and applications in network coding and related fields.

Contribution

It establishes that results and proofs for rank weights and Hamming weights are essentially equivalent, revealing new insights and consequences for both in network coding.

Findings

Many results in Hamming weights have rank weight analogs

Results and proofs in both cases are fundamentally similar

New consequences for Hamming weights derived from rank case insights

Abstract

Rank weights and generalized rank weights have been proven 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 cases have been established and proven 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 proven 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.