On Circular-Shift Linear Solvability of Multicast Networks

Circular-shift linear network coding (LNC) is a class of LNC schemes with low coding complexities. However, there are explicit multicast networks whose capacities cannot be achieved by circular-shift LNC. In this work, we first extend the formulation of circular-shift LNC from over GF(2) to over GF(<inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>), where <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> is an arbitrary prime. Then, in terms of scalar linear solvability, we characterize an equivalent condition on the circular-shift linear solvability of an arbitrary multicast network. Specifically, we prove that a multicast network has a circular-shift linear solution over GF(<inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>) if and only if it has a scalar linear solution over GF(<inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>). This implies that the study of circular-shift LNC at rate smaller than 1 is inevitable. We last prove that every multicast network is asymptotically circular-shift linearly solvable over GF(<inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>), that is, for any <inline-formula> <tex-math notation="LaTeX">$\epsilon > 0$ </tex-math></inline-formula>, it has a circular-shift linear solution over GF(<inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>) at rate larger than <inline-formula> <tex-math notation="LaTeX">$1 - \epsilon $ </tex-math></inline-formula>.