This preview shows page 1. Sign up to view the full content.
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...
View Full Document
- Fall '13