Characteristic Sets of Fixed-Dimension Vector Linear Codes for Non-Multicast Networks

Vector linear solvability of non-multicast networks depends upon both the characteristic of the finite field and the dimension of the vector linear network code. In the literature, the dependency on the characteristic of the finite field and the dependency on the dimension have been studied separately. In this paper, we show the interdependency between the characteristic of the finite field and the dimension of the vector linear network code that achieves a vector linear network coding (VLNC) solution in non-multicast networks. For any given network <inline-formula> <tex-math notation="LaTeX">$\mathcal {N}$ </tex-math></inline-formula>, we define <inline-formula> <tex-math notation="LaTeX">$P(\mathcal {N},d)$ </tex-math></inline-formula> as the set of all characteristics of finite fields over which the network <inline-formula> <tex-math notation="LaTeX">$\mathcal {N}$ </tex-math></inline-formula> has a <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>-dimensional VLNC solution. To the best of our knowledge, for any network <inline-formula> <tex-math notation="LaTeX">$\mathcal {N}$ </tex-math></inline-formula> shown in the literature, if <inline-formula> <tex-math notation="LaTeX">$P(\mathcal {N},1)$ </tex-math></inline-formula> is non-empty, then <inline-formula> <tex-math notation="LaTeX">$P(\mathcal {N},1) = P(\mathcal {N},d)$ </tex-math></inline-formula> for any positive integer <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>. We show that, for any two non-empty sets of primes <inline-formula> <tex-math notation="LaTeX">$P_{1}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$P_{2}$ </tex-math></inline-formula>, there exists a network <inline-formula> <tex-math notation="LaTeX">$\mathcal {N}$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$P(\mathcal {N},1) = P_{1}$ </tex-math></inline-formula>, but <inline-formula> <tex-math notation="LaTeX">$P(\mathcal {N},2) = \{P_{1},P_{2} \}$ </tex-math></inline-formula>. We also show that there are networks exhibiting a similar advantage (the existence of a VLNC solution over a larger set of characteristics) if the dimension is increased from 2 to 3. However, such behaviour is not universal, as there exist networks which admit a VLNC solution over a smaller set of characteristics of finite fields when the dimension is increased. Using the networks constructed in this paper, we further demonstrate that: (i) a network having an <inline-formula> <tex-math notation="LaTeX">$m_{1}$ </tex-math></inline-formula>-dimensional VLNC solution over a finite field of some characteristic and an <inline-formula> <tex-math notation="LaTeX">$m_{2}$ </tex-math></inline-formula>-dimensional VLNC solution over a finite field of some other characteristic may not have an <inline-formula> <tex-math notation="LaTeX">$(m_{1} + m_{2})$ </tex-math></inline-formula>-dimensional VLNC solution over any finite field; (ii) there exist a class of networks for which scalar linear network coding (SLNC) over non-commutative rings has some advantage over SLNC over finite fields: the least sized non-commutative ring over which each network in the class has an SLNC solution is significantly lesser in size than the least sized finite field over which it has an SLNC solution.