Multicast Network Coding and Field Sizes

In an acyclic multicast network, it is well known that a linear network coding solution over GF($q$) exists when $q$ is sufficiently large. In particular, for each prime power $q$ no smaller than the number of receivers, a linear solution over GF($q$) can be efficiently constructed. In this work, we reveal that a linear solution over a given finite field does \emph{not} necessarily imply the existence of a linear solution over all larger finite fields. Specifically, we prove by construction that: (i) For every source dimension no smaller than 3, there is a multicast network linearly solvable over GF(7) but not over GF(8), and another multicast network linearly solvable over GF(16) but not over GF(17); (ii) There is a multicast network linearly solvable over GF(5) but not over such GF($q$) that $q > 5$ is a Mersenne prime plus 1, which can be extremely large; (iii) A multicast network linearly solvable over GF($q^{m_1}$) and over GF($q^{m_2}$) is \emph{not} necessarily linearly solvable over GF($q^{m_1+m_2}$); (iv) There exists a class of multicast networks with a set $T$ of receivers such that the minimum field size $q_{min}$ for a linear solution over GF($q_{min}$) is lower bounded by $\Theta(\sqrt{|T|})$, but not every larger field than GF($q_{min}$) suffices to yield a linear solution. The insight brought from this work is that not only the field size, but also the order of subgroups in the multiplicative group of a finite field affects the linear solvability of a multicast network

Turbulent entrainment origin of protostellar outflows

Protostellar outflow is a prominent process that accompanies the formation of stars. It is generally agreed that wide-angled protostellar outflows come from the interaction between the wind from a forming star and the ambient gas. However, it is still unclear how the interaction takes place. In this work, we theoretically investigate the possibility that the outflow results from interaction between the wind and the ambient gas in the form of turbulent entrainment. In contrast to the previous models, turbulent motion of the ambient gas around the protostar is taken into account. In our model, the ram-pressure of the wind balances the turbulent ram-pressure of the ambient gas, and the outflow consists of the ambient gas entrained by the wind. The calculated outflow from our modelling exhibits a conical shape. The total mass of the outflow is determined by the turbulent velocity of the envelope as well as the outflow age, and the velocity of the outflow is several times higher than the velocity dispersion of the ambient gas. The outflow opening angle increases with the strength of the wind and decreases with the increasing ambient gas turbulence. The outflow exhibits a broad line width at every position. We propose that the turbulent entrainment process, which happens ubiquitously in nature, plays a universal role in shaping protostellar outflows.Comment: 15 pages, accepted for publication in A&