Abstract-This paper investigates the construction of linear network codes for broadcasting a set of data packets to a number of users. The links from the source to the users are modeled as independent erasure channels. Users are allowed to inform the source node whether a packet is received correctly via feedback channels. In order to minimize the number of packet transmissions until all users have received all packets successfully, it is necessary that a data packet, if successfully received by a user, can increase the dimension of the vector space spanned by the encoding vectors he or she has received by one. Such an encoding vector is called innovative. To reduce decoding complexity, sparse encoding vectors are preferred, since the sparsity can be exploited when solving systems of linear equations. Generating a sparsest encoding vector with large finite field size, however, is shown to be NP-hard. An approximation algorithm is constructed. For binary field, heuristic algorithms are also proposed. Index Terms-Erasure broadcast channel, network coding, computational complexity
