of linear codes, conditions to be rank degenerate, duality properties and MRD ranks.\
author:
- 'Umberto Mart[í]{}nez-Peñas,  [^1]'
title: 'Generalized rank weights of reducible codes, optimal cases and related properties'
---

[Shell : Bare Demo of IEEEtran.cls for Journals]{}

Generalized rank weight, rank-metric codes, rank distance, rank equivalent codes, reducible codes, secure network coding.

Introduction
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network coding was first studied in [@ahlswede; @linearnetwork], further formalized in [@Koetter2003], and provides higher throughput than storing and forwarding messages on the network. Two of the *main problems* in this context are error and erasure correction, and security against information leakage to a wire-tapper, which were first studied in [@cai-yeung] and [@secure-network], respectively.

Rank-metric codes were found to be universally suitable (meaning independently of the underlying network code) for error and erasure correction in linear network coding in [@on-metrics], used as forward error-correcting codes, and they were found to be universally suitable against information leakage in [@silva-universal], used in the form of coset coding. Both constructions can be treated separately and applied together in a concatenated way (see [@silva-universal Sec. VII-B]).

On the security side, generalized rank weights (GRWs) of codes that are linear over the extension field were introduced in [@rgrw; @oggier] 