This paper addresses the problem of multicasting in directed acyclic graphs (DAGs) to maximize the data rate. It demonstrates that coding at intermediate nodes can significantly increase the achievable rate compared to uncoded multicasting. The authors provide deterministic and randomized polynomial-time algorithms for designing linear codes for directed acyclic graphs with unit-capacity edges. These algorithms are extended to handle integer capacities and robust codes that can tolerate edge failures. The main results include: 1. **Polynomial-Time Algorithms**: deterministic and randomized algorithms for designing linear codes for directed acyclic graphs with unit-capacity edges. 2. **Integer Capacities**: algorithms that handle integer edge capacities, reducing the maximum flow to a polynomial-time problem. 3. **Robust Codes**: algorithms that construct robust linear network codes that can tolerate edge failures, with a finite field size that is exponentially smaller than the number of sinks. The paper also discusses the gap between the achievable rates with and without coding, showing that coding can increase the rate by a factor of \(\Omega(\log |V|)\) in certain networks. The algorithms are designed to operate over finite fields of size \(2^m\), where \(m\) is the number of bits per symbol, and the resulting codes are much smaller than those in previous constructions. The paper concludes with a discussion of open problems, including the complexity of handling capacitated edges and the more general network coding problem involving multiple senders and receivers.This paper addresses the problem of multicasting in directed acyclic graphs (DAGs) to maximize the data rate. It demonstrates that coding at intermediate nodes can significantly increase the achievable rate compared to uncoded multicasting. The authors provide deterministic and randomized polynomial-time algorithms for designing linear codes for directed acyclic graphs with unit-capacity edges. These algorithms are extended to handle integer capacities and robust codes that can tolerate edge failures. The main results include: 1. **Polynomial-Time Algorithms**: deterministic and randomized algorithms for designing linear codes for directed acyclic graphs with unit-capacity edges. 2. **Integer Capacities**: algorithms that handle integer edge capacities, reducing the maximum flow to a polynomial-time problem. 3. **Robust Codes**: algorithms that construct robust linear network codes that can tolerate edge failures, with a finite field size that is exponentially smaller than the number of sinks. The paper also discusses the gap between the achievable rates with and without coding, showing that coding can increase the rate by a factor of \(\Omega(\log |V|)\) in certain networks. The algorithms are designed to operate over finite fields of size \(2^m\), where \(m\) is the number of bits per symbol, and the resulting codes are much smaller than those in previous constructions. The paper concludes with a discussion of open problems, including the complexity of handling capacitated edges and the more general network coding problem involving multiple senders and receivers.