Linear Operator Broadcast Channels

Abstract---Recent research indicates that packet multicasting employing random lin-ear network coding can be regarded as transmitting subspaces as symbols over a linear operator channel (LOC), a discrete memoryless unicast subspace channel over some finite field. This thesis addresses the problem of packet broadcast over linear operator channels, it is an important problem to consider, since currently most of the existing literature focuses on the easier multicast problem, whereas the problem of characterizing good schemes to transmit to receivers with different min-cuts is open. The work constitutes of three parts. In the first part, we propose the framework of linear operator broadcast channels (LOBCs) and start initial work on determining their capacity region. Our discussion concerns a class of constant-dimension multiplicative LOBCs (CMLOBCs), whose input subspaces have constant dimension and missing vectors of a subspace constitute the only possible channel interference. CMLOBC can be viewed as a generalization of broadcast erasure channel. Two fundamental problem regarding CMLOBCs are addressed--the necessary and suffi-cient condition on CMLOBC being stochastically degraded and whether time sharing is suffice to achieve its capacity region in degraded case. In the second part, we setup the framework of algebraic coding theory over LOBCs and prove an upper bound on broad-cast subspace codes with respect to separation vector under subspace distance metric. The broadcast subspace codes inherit the principle of unequal error protection codes. Finally, we construct broadcast subspace codes by multilevel construction. We use linear block code to modulate the subspace symbols on projective space as what we used to do for signal symbols on Euclidean plane. By the nature of multilevel construction, the UEP property can be obtained with ease.