The Multicast Solvability of Permutation Linear Network Coding

Permutation linear network coding (LNC) is a generalization of circular-shift LNC. It has a much richer supply of linear coding operations which can be efficiently implemented. It is known that circular-shift LNC is insufficient to achieve the exact capacity of certain multicast networks. It would be natural to ask whether permutation LNC can achieve the exact capacity of every multicast network. In this letter, we prove that a multicast network has an $L$ -dimensional permutation linear solution over a ring $\mathbb {R}$ if and only if it has a scalar linear solution over $\mathbb {R}$ . This result implies that the capacity of a multicast network not scalar linearly solvable over $\mathbb {R}$ cannot be exactly achieved by permutation LNC either. On the other hand, we unveil, by an explicit instance, the advantage of permutation LNC over circular-shift LNC in terms of the shorter block length to yield a linear solution at rate smaller than 1.