See Exercise 53 for some background. When information is transmitted, there may be some errors in the communication. We present a method of adding extra information to messages so that most errors that occur during transmission can be detected and corrected. Such methods are referred to as error-correcting codes. (Compare these with codes whose purpose is to conceal information.) The pictures of man’s first landing on the Moon (in 1969) were televised just as they had been received and were not very clear, since they contained many errors induced during transmission. On later missions, much clearer error-corrected pictures were obtained.
In computers, information is stored and processed in the form of strings of binary digits, 0 and 1. This stream of binary digits is often broken up into “blocks” of eight binary digits (bytes). For the sake of simplicity, we will work with blocks of only four binary digits (i.e., with
Suppose these vectors in
a. Let H he the Hamming matrix introduced in Exercise 53. How can the computer in Kourou use
b. Suppose the vector
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