On polarization for the linear operator channel

We address the problem of reliably transmitting information through a network where the nodes perform random linear network coding and where an adversary potentially injects malicious packets into the network. A good model for such a channel is a linear operator channel, where in this work we employ a combined multiplicative and additive matrix channel. We show that this adversarial channel behaves like a subspace-based symmetric discrete memoryless channel (DMC) under subspace insertions and deletions and typically has an input alphabet with non-prime cardinality. This facilitates the recent application of channel polarization results for DMCs with arbitrary input alphabets by providing a suitable one-to-one mapping from input matrices to subspaces. As a consequence, we show that polarization for this adversarial linear operator channel can be obtained via an element-wise encoder mapping for the input matrices, which replaces the finite field summation in the channel combining step for Arikan's classical polar codes.