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

Low Complexity Encoding for Network Codes

In this paper we consider the per-node run-time complexity of network multicast codes. We show that the randomized algebraic network code design algorithms described extensively in the literature result in codes that on average require a number of operations that scales quadratically with the blocklength m of the codes. We then propose an alternative type of linear network code whose complexity scales linearly in m and still enjoys the attractive properties of random algebraic network codes. We also show that these codes are optimal in the sense that any rate-optimal linear network code must have at least a linear scaling in run-time complexity

Hamming weights and Betti numbers of Stanley-Reisner rings associated to matroids

To each linear code over a finite field we associate the matroid of its parity check matrix. We show to what extent one can determine the generalized Hamming weights of the code (or defined for a matroid in general) from various sets of Betti numbers of Stanley-Reisner rings of simplicial complexes associated to the matroid

Zero forcing in iterated line digraphs

Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks. In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications

A decoding algorithm for Twisted Gabidulin codes

In this work, we modify the decoding algorithm for subspace codes by Koetter and Kschischang to get a decoding algorithm for (generalized) twisted Gabidulin codes. The decoding algorithm we present applies to cases where the code is linear over the base field $\mathbb{F}_q$ but not linear over $\mathbb{F}_{q^n}$.Comment: This paper was submitted to ISIT 201

MIMO Multiway Relaying with Clustered Full Data Exchange: Signal Space Alignment and Degrees of Freedom

We investigate achievable degrees of freedom (DoF) for a multiple-input multiple-output (MIMO) multiway relay channel (mRC) with $L$ clusters and $K$ users per cluster. Each user is equipped with $M$ antennas and the relay with $N$ antennas. We assume a new data exchange model, termed \emph{clustered full data exchange}, i.e., each user in a cluster wants to learn the messages of all the other users in the same cluster. Novel signal alignment techniques are developed to systematically construct the beamforming matrices at the users and the relay for efficient physical-layer network coding. Based on that, we derive an achievable DoF of the MIMO mRC with an arbitrary network configuration of $L$ and $K$, as well as with an arbitrary antenna configuration of $M$ and $N$. We show that our proposed scheme achieves the DoF capacity when $\frac{M}{N} \leq \frac{1}{LK-1}$ and $\frac{M}{N} \geq \frac{(K-1)L+1}{KL}$.Comment: 13 pages, 4 figure