On Vector Linear Solvability of Multicast Networks.

Vector linear network coding (LNC) is a generalization of the conventional scalar LNC, such that the data unit transmitted on every edge is an L -dimensional vector of data symbols over a base field GF( q ). Vector LNC enriches the choices of coding operations at intermediate nodes, and there is a popular conjecture on the benefit of vector LNC over scalar LNC in terms of alphabet size of data units: there exist (single-source) multicast networks that are vector linearly solvable of dimension L over GF( q . This paper introduces a systematic way to construct such multicast networks, and subsequently establish explicit instances to affirm the positive answer of this conjecture for infinitely many alphabet sizes p^{L} with respect to an arbitrary prime p$ . On the other hand, this paper also presents explicit instances with the special property that they do not have a vector linear solution of dimension L$ over GF(2) but have scalar linear solutions over GF( q'$ ) for some , where $q'$ can be odd or even. This discovery also unveils that over a given base field, a multicast network that has a vector linear solution of dimension $L$ does not necessarily have a vector linear solution of dimension $L' > L$ . [ABSTRACT FROM AUTHOR]