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Vector linear network coding (LNC) is a generalization of the conventional scalar LNC, such that the data unit transmitted on every edge is an $L$-dimensional vector of data symbols over a base field GF($q$). Vector LNC enriches the choices of coding operations at intermediate nodes, and there is a popular conjecture on the benefit of vector LNC over scalar LNC in terms of alphabet size of data units: there exist (single-source) multicast networks that are vector linearly solvable of dimension $L$ over GF($q$) but not scalar linearly solvable over any field of size $q' \leq q^L$. This paper introduces a systematic way to construct such multicast networks, and subsequently establish explicit instances to affirm the positive answer of this conjecture for \emph{infinitely many} alphabet sizes $p^L$ with respect to an \emph{arbitrary} prime $p$. On the other hand, this paper also presents explicit instances with the special property that they do not have a vector linear solution of dimension $L$ over GF(2) but have scalar linear solutions over GF($q'$) for some $q' < 2^L$, where $q'$ can be odd or even. This discovery also unveils that over a given base field, a multicast network that has a vector linear solution of dimension $L$ does not necessarily have a vector linear solution of dimension $L' > L$.

Distributed storage systems provide large-scale reliable data storage services by spreading redundancy across a large group of storage nodes. In such a large system, node failures take place on a regular basis. When a storage node breaks down, a replacement node is expected to regenerate the redundant data as soon as possible in order to maintain the same level of redundancy. Previous results have been mainly focused on the minimization of network traffic in regeneration. However, in practical networks, where link capacities vary in a wide range, minimizing network traffic does not always yield the minimum regeneration time. In this paper, we investigate two approaches to the problem of minimizing regeneration time in networks with heterogeneous link capacities. The first approach is to download different amounts of repair data from the helping nodes according to the link capacities. The second approach generalizes the conventional star-structured regeneration topology to tree-structured topologies so that we can utilize the links between helping nodes with bypassing low-capacity links. Simulation results show that the flexible tree-structured regeneration scheme that combines the advantages of both approaches can achieve a substantial reduction in the regeneration time., Comment: submitted to Trans. IT in Feb. 2016

In an acyclic multicast network, it is well known that a linear network coding solution over GF($q$) exists when $q$ is sufficiently large. In particular, for each prime power $q$ no smaller than the number of receivers, a linear solution over GF($q$) can be efficiently constructed. In this work, we reveal that a linear solution over a given finite field does \emph{not} necessarily imply the existence of a linear solution over all larger finite fields. Specifically, we prove by construction that: (i) For every source dimension no smaller than 3, there is a multicast network linearly solvable over GF(7) but not over GF(8), and another multicast network linearly solvable over GF(16) but not over GF(17); (ii) There is a multicast network linearly solvable over GF(5) but not over such GF($q$) that $q > 5$ is a Mersenne prime plus 1, which can be extremely large; (iii) A multicast network linearly solvable over GF($q^{m_1}$) and over GF($q^{m_2}$) is \emph{not} necessarily linearly solvable over GF($q^{m_1+m_2}$); (iv) There exists a class of multicast networks with a set $T$ of receivers such that the minimum field size $q_{min}$ for a linear solution over GF($q_{min}$) is lower bounded by $\Theta(\sqrt{|T|})$, but not every larger field than GF($q_{min}$) suffices to yield a linear solution. The insight brought from this work is that not only the field size, but also the order of subgroups in the multiplicative group of a finite field affects the linear solvability of a multicast network.

As storage systems grow in size, device failures happen more frequently than ever before. Given the commodity nature of hard drives employed, a storage system needs to tolerate a certain number of disk failures while maintaining data integrity, and to recover lost data with minimal interference to normal disk I/O operations. RAID-6, which can tolerate up to two disk failures with the minimum redundancy, is becoming widespread. However, traditional RAID-6 codes suffer from high disk I/O overhead during recovery. In this paper, we propose a new family of RAID-6 codes, the Minimum Disk I/O Repairable (MDR) codes, which achieve the optimal disk I/O overhead for single failure recoveries. Moreover, we show that MDR codes can be encoded with the minimum number of bit-wise XOR operations. Simulation results show that MDR codes help to save about half of disk read operations than traditional RAID-6 codes, and thus can reduce the recovery time by up to 40%., Comment: Accepted version. Please refer to http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6804945 for the published version. 0733-8716/14/$31.00 \c{opyright} 2014 IEEE

Network Coding encourages information coding across a communication network. While the necessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often treats the network topology as a black box, focusing on algebraic or information theoretic aspects of the problem. This work aims at an in-depth examination of the relation between algebraic coding and network topologies. We mathematically establish a series of results along the direction of: if network coding is necessary/beneficial, or if a particular finite field is required for coding, then the network must have a corresponding hidden structure embedded in its underlying topology, and such embedding is computationally efficient to verify. Specifically, we first formulate a meta-conjecture, the NC-Minor Conjecture, that articulates such a connection between graph theory and network coding, in the language of graph minors. We next prove that the NC-Minor Conjecture is almost equivalent to the Hadwiger Conjecture, which connects graph minors with graph coloring. Such equivalence implies the existence of $K_4$, $K_5$, $K_6$, and $K_{O(q/\log{q})}$ minors, for networks requiring $\mathbb{F}_3$, $\mathbb{F}_4$, $\mathbb{F}_5$ and $\mathbb{F}_q$, respectively. We finally prove that network coding can make a difference from routing only if the network contains a $K_4$ minor, and this minor containment result is tight. Practical implications of the above results are discussed., Comment: 26 pages, 12 (sub-)figures, partially presented in INFOCOM 2013, submitted to Trans. IT