The overhead of the replication code of rate c is 1

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Unformatted text preview: c total bits. The overhead of the replication 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 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 and parity calculations in the context of two classes of codes: linear block codes, which are an instance of the broad class of algebraic codes, and convolutional codes, which are an instance of the broad class of...
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