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) but not scalar linearly solvable over any field of size q ′ ≤ q L . 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 q ′ < 2 L , 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 Vector network coding, scalar network coding, multicast networks, alphabet size, direct sum. The preliminary results of this paper were partially presented at
al REF demonstrated that, while vector linear codes can outperform scalar linear codes in terms yielding solutions for general networks, there can exist multicast networks that are not k-dimensional vector linearly solvable over GF(2) yet have scalar linear solutions over some field alphabet whose size is less than 2 k .
On Vector Linear Solvability of Multicast Networks
IEEE TRANSACTIONS ON COMMUNICATIONS, vol. 64, no. 12, pp. 5096-5107, 2016
Mathematics Computer Science