Bridging Shannon and Hamming: Codes for computationally simple channels
Speaker: Venkatesan Guruswami , Carnegie Mellon U.
Date: March 8 2011
Time: 4:15PM to 5:15PM
Location: 32-155
Host: Scott Aaronson, CSAIL, MIT
Contact: Be Blackburn , 3-6098, imbe@mit.edu
Relevant URL: The theory of error-correcting codes has had two divergent schools of
thought, dating back to its origins, based on the underlying model of
the noisy channel. Shannon's theory modeled the channel as a
stochastic process with a known probability law. Hamming's work
suggested a worst-case model, where the channel is subject only to a
limit on the number of errors it may cause. These two approaches share
several common tools, however in terms of quantitative results, the
classical results in the harsher Hamming model are much weaker.
In this talk, we will discuss a line of research aimed at bridging
between these models. We will begin by surveying some approaches that
rely on setup assumptions (such as shared randomness) to construct
codes against worst-case errors with information rate similar to what
is possible against random errors. We then turn to our results for
computationally bounded channels, which can introduce an arbitrary set
of errors as long as (a) the total fraction of errors is bounded by a
parameter p, and (b) the process which adds the errors is sufficiently
``simple'' computationally. Such channel models are well-motivated
since physical noise processes may be mercurial, but are not
computationally intensive. Also, as with codes for worst-case errors,
codes for such channels can handle errors whose true behavior is
unknown or varying over time.
We will describe an explicit construction of a poly-time
encodable/decodable code of rate approaching Shannon capacity 1-h(p)
that can correct an arbitrary error pattern with total fraction of
errors bounded by p, provided the channel's action is oblivious to the
codeword. We will hint at an extension to channels limited to online
logarithmic space that gives efficient codes with optimal rate that
enable recovery of a short list containing the correct message. (A
similar claim holds for channels admitting polynomial size circuits,
assuming the existence of pseudorandom generators.) Our results do not
use any shared randomness or other setup assumptions.
Based on joint work with Adam Smith.
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