Error-correction coding for an operator channel: Combinatorial bounds on the trade-off between the size of these codes and their minimum distance

Meeting Time: Oct. 6, 2009, 2:00-2:50pm

Abstract: We will examine a new channel model that has arisen in the context of network coding. This model, known as an operator channel, takes subspaces as inputs to the channel. These subspaces then undergo some errors and erasures during transmission, which are modeled as a loss of dimensions of the subspace and as the addition of a subspace of errors, and the transformed subspaces are then received as outputs of the channel. Given this new channel model, the task is then to design good codes for this channel, i.e. sets of subspaces of (F_q)^N that maximize the trade-off between the number of codewords and the minimum distance between codewords.A number of combinatorial bounds describing this trade-off will be proven that are analogous to the fundamental sphere-packing, sphere-covering, and Singleton bounds of classical block coding. Additionally, we will examine the construction of a family of subspace codes that achieve one of these bounds.