On the Dependence of Linear Coding Rates on the Characteristic of the Finite Field

It is known that for any finite/co-finite set of primes there exists a network which has a rate $1$ solution if and only if the characteristic of the finite field belongs to the given set. We generalize this result to show that for any positive rational number $k/n$, and for any given finite/co-finite set of primes, there exists a network which has a rate $k/n$ fractional linear network coding solution if and only if the characteristic of the finite field belongs to the given set. For this purpose we construct two networks: $\mathcal{N}_1$ and $\mathcal{N}_2$; the network $\mathcal{N}_1$ has a $k/n$ fractional linear network coding solution if and only if the characteristic of the finite field belongs to the given finite set of primes, and the network $\mathcal{N}_2$ has a $k/n$ fractional linear network coding solution if and only if the characteristic of the finite field belongs to the given co-finite set of primes. Recently, a method has been introduced where characteristic-dependent linear rank inequalities are produced from networks whose linear coding capacity depends on the characteristic of the finite field. By employing this method on the networks $\mathcal{N}_1$ and $\mathcal{N}_2$, we construct two classes of characteristic-dependent linear rank inequalities. For any given set of primes, the first class contains an inequality which holds if the characteristic of the finite field does not belong to the given set of primes but may not hold otherwise; the second class contains an inequality which holds if the characteristic of the finite field belongs to the given set of primes but may not hold otherwise. We then use these inequalities to obtain an upper-bound on the linear coding capacity of $\mathcal{N}_1$ and $\mathcal{N}_2$.