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L6_2 - MIT 6.02 DRAFT Lecture Notes Fall 2010(Last update...

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MIT 6.02 DRAFT Lecture Notes Fall 2010 (Last update: October 7, 2010) Comments, questions or bug reports? Please contact [email protected] L ECTURE 6 Coping with Bit Errors Recall our main goal in designing digital communication networks: to send information both reliably and efficiently between nodes. Meeting that goal requires the use of tech- niques to combat bit errors, which are an inevitable property of both commmunication channels and storage media. The key idea we will apply to achieve reliable communication is redundancy : by repli- cating data to ensure that it can be reconstructed from some (correct) subset of the data received, we can improve the likelihood of fixing errors. This approach, called error cor- rection , can work quite well when errors occur according to some random process. Error correction may not be able to correct all errors, however, so we will need a way to tell if any errors remain. That task is called error detection , which determines whether the data received after error correction is in fact the data that was sent. It will turn out that all the guarantees we can make are probabilistic, but we will be able to make our guarantees on reliabile data receptions with very high probability. Over a communication channel, the sender and receiver implement the error correction and detection procedures. The sender has an encoder whose job is to take the message and process it to produce the coded bits that are then sent over the channel. The receiver has a decoder whose job is to take the re- ceived (coded) bits and to produce its best estimate of the message. The encoder-decoder procedures together constitute channel coding . Our plan is as follows. First, we will define a model for the kinds of bit errors we’re going to handle and revisit the previous lectures to see why errors occur. Then, we will discuss and analyze a simple redundancy scheme called a replication code , which will sim- ply make c copies of any given bit. The replication code has a code rate of 1 /c —that is, for every useful bit we receive, we will end up encoding c total bits. The overhead of the repli- cation code of rate c is 1 1 /c , which is rather high for the error correcting power of the code. We will then turn to the key ideas in that allow us to build powerful codes capable of correcting errors without such a high overhead (or, capable of correcting far more errors at a given code rate than the trivial replication code). There are two big ideas that are used in essentially all channel codes: the first is the notion of embedding , where the messages one wishes to send are placed in a geometrically pleasing way in a larger space so that the distance between any two valid points in the 1
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2 LECTURE 6. COPING WITH BIT ERRORS embedding is large enough to enable the correction and detection of errors. The second big idea is to use parity calculations (or more generally, linear functions) over the bits we wish to send to produce the bits that are actually sent. We will study examples of embeddings
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