Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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($q$) but not scalar
linearly solvable over any field of size $q' \leq q^L$. 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
\emph{infinitely many} alphabet sizes $p^L$ with respect to an \emph{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 linear solution of
dimension $L' > L$.

13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
19
The categories of this paper are: eess
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int