Suppose we are trying to send a series of bits through a channel with a bit erro

Question

Suppose we are trying to send a series of bits through a channel with a bit error rate (i.e., each bit has
probability of being received incorrectly). One way to improve the reliability is to use repetition code (3,1).
For example, if we want to send the bits `011', we actually send `000111111'. The receiver looks at the bits
in groups of three, and decodes each group to the bit that occurs most often in the group. What is the error
rate by using repetition code (3,1) (i.e., the probability that a group of three repeated bits will be decoded
incorrectly given )? How much does this repetition code improve reliability?

Explanation / Answer

Repeating each bit three times allows us to correct one error in each group of three bits but not more errors.

Letvus consider each bit has probability of P of being recieved correctly,independently for each bit. Then probability that a group of three repeated bits will be decoded is:

Pr(0 errors)+ Pr(1 error)=P3+3P2(1-p)

Suppose we want an odd number, n, times then we will be able to decode correctly as long as there are less tan n/2 errors.
As n gets bigger the decoding error probability goes down.

Number of repetitions probability
3 0.028
5 0.0086
7 0.0027
9 0.00089

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