We can also easily generalize this idea to check if a

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Unformatted text preview: + 1 errors that cannot be corrected reliably. Equation (6.4) gives us a way of determining if single-bit error correction can always be performed on a proposed set S of transmission messages—we could write a program to compute the Hamming distance between all pairs of messages in S and verify that the minimum Hamming distance was at least 3. We can also easily generalize this idea to check if a code can always correct more errors. And we can use the observations made above to decode any received word: just find the closest valid code word to the received one, and then use the known mapping between each distinct message and the code word to produce the message. That check may be exponential in the number of message bits we would like to send, but would be reasonable if the number of bits is small. But how do we go about finding a good embedding (i.e., good code words)? This task isn’t straightforward, as the following example shows. Suppose we want to reliably send 4-bit messages so...
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