Abstract-The famous max-flow min-cut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the min-cut separating and . Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures.

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Recently there has been interest in the construction of small parity check sets for iterative decoding of the Hamming code with the property that each uncorrectable (or stopping) set of size three is the support of a codeword and hence uncorrectable anyway. Here we reformulate and generalise the problem and improve on this construction.First we show that a parity check collection that corrects all correctable erasure patterns of size m for the r-th order Hamming code (i.e, the Hamming code with codimension r) provides for all codes of codimension r a corresponding "generic" parity check collection with this property. This leads naturally to a necessary and sufficient condition on such generic parity check collections. We use this condition to construct a generic parity check collection for codes of codimension r correcting all correctable erasure patterns of size at most m, for all r and m ≤ r, thus generalising the known construction for m = 3. Then we discuss optimality of our construction and show that it can be improved for m ≥ 3 and r large enough. Finally we discuss some directions for further research. * The authors are with