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We develop new algebraic algorithms for scalar and vector network coding. In vector network coding, the source multicasts information by transmitting vectors of length L, while intermediate nodes process and combine their incoming packets by multiplying them with L x L coding matrices that play a similar role as coding coefficients in scalar coding. We start our work by extending the algebraic framework developed for multicasting over graphs by Koetter and Medard to include operations over matrices; we build on this generalized framework, to provide a new approach for both scalar and vector code design which attempts to minimize the employed field size and employed vector length, while selecting the coding operations. Our algorithms also lead as a special case to network code designs that employ structured matrices.