Lets dene one more piece of notation let ewi be the

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Unformatted text preview: n extend our approach by producing an embedding with more space between valid codewords! Suppose we limit our selection of messages in S even further, as follows: HD(wi , wj ) ≥ 3 for all wi , wj ∈ S where i ￿= j (6.4) How does it help to increase the minimum Hamming distance to 3? Let’s define one more piece of notation: let Ewi be the set of messages resulting from corrupting wi with a single error. For example, Eo00 = {001, 010, 100}. Note that HD(wi , an element of Ewi ) = 1. With a minimum Hamming distance of 3 between the valid code words, observe that there is no intersection between Ewi and Ewj when i ￿= j . Why is that? Suppose there was a message wk that was in both Ewi and Ewj . We know that HD(wi , wk ) = 1 and HD(wj , wk ) = 1, which implies that wi and wj differ in at most two bits and consequently HD(wi , wj ) ≤ 2. That contradicts our specification that their minimum Hamming distance be 3. So the Ewi don’t intersect. Now we can correct single bit errors as well: the received m...
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