Methods for Efficient Network Coding

I. INTRODUCTION Random linear network coding is a multicast communication scheme in which all participating nodes send out coded packets formed from random linear combinations of packets received so far. This scheme is capacity-achieving for single sessions over lossy wireline or wireless packet networks [1], [2], [3], [4]. Thus, from the point of view of efficiently utilizing transmissions, random linear network coding is an attractive strategy. However, it is not presently attractive from the point of view of efficiently utilizing computational resources. To decode a k-packet-long message, a decoder needs to invert a k×k dense matrix, which, using Gaussian elimination, requires O(k 3) operations (or O(k 2) operations per input symbol). Once the matrix inverse is computed, applying the inverse to the received coded packets to recover the message requires O(k 2) operations (or O(k) operations per input symbol). Although the former computation is more costly than the latter as a function of k, it is usually the latter cost that dominates, since the latter computation depends on the length of the packets (which is usually on the order of kilobytes), while the former does not. This dominant cost can make the computational resources required for random linear network coding prohibitive [5]. But a random linear code is a somewhat na¨ıve code. By picking the code randomly, we ensure that the code is efficient at communicating information but, at the same time, by employing little design in the code, we obtain a code that is computationally inefficient. Thus, a natural code design question arises: can we design a network code that preserves the communication efficiency of a random linear code and achieves better computational efficiency? In this paper, we give an affirmative answer to this question. The code that we present as our answer achieves significantly better computational efficiency and is based primarily on techniques that are now quite standard in the literature on erasure codes. That said, our application of these techniques to network coding requires some novel ideas. First, we partition