Bounds for flag codes

The application of flags to network coding has been introduced recently by Liebhold, Nebe, and Vazquez-Castro. It is a variant to random linear network coding and explicit routing solutions for given networks. Here we study lower and upper bounds for the maximum possible cardinality of a corresponding flag code with given parameters.Comment: 23 pages, 6 tables, typos correcte

End-to-End Algebraic Network Coding for Wireless TCP/IP Networks

The Transmission Control Protocol (TCP) was designed to provide reliable transport services in wired networks. In such networks, packet losses mainly occur due to congestion. Hence, TCP was designed to apply congestion avoidance techniques to cope with packet losses. Nowadays, TCP is also utilized in wireless networks where, besides congestion, numerous other reasons for packet losses exist. This results in reduced throughput and increased transmission round-trip time when the state of the wireless channel is bad. We propose a new network layer, that transparently sits below the transport layer and hides non congestion-imposed packet losses from TCP. The network coding in this new layer is based on the well-known class of Maximum Distance Separable (MDS) codes.Comment: Accepted for the 17th International Conference on Telecommunications 2010 (ICT2010), Doha, Qatar, April 4 - 7, 2010. 6 pages, 7 figure

Maximum Flag-Rank Distance Codes

In this paper we extend the study of linear spaces of upper triangular matrices endowed with the flag-rank metric. Such metric spaces are isometric to certain spaces of degenerate flags and have been suggested as suitable framework for network coding. In this setting we provide a Singleton-like bound which relates the parameters of a flag-rank-metric code. This allows us to introduce the family of maximum flag-rank distance codes, that are flag-rank-metric codes meeting the Singleton-like bound with equality. Finally, we provide several constructions of maximum flag-rank distance codes

A Polymatroid Approach to Generalized Weights of Rank Metric Codes

We consider the notion of a $(q,m)$-polymatroid, due to Shiromoto, and the more general notion of $(q,m)$-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martinez-Penas and Matsumoto for relative generalized rank weights are derived as a consequence.Comment: 22 pages; with minor revisions in the previous versio