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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.