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Concept
Repetition code
Summary
In coding theory, the repetition code is one of the most basic linear error-correcting codes. In order to transmit a message over a noisy channel that may corrupt the transmission in a few places, the idea of the repetition code is to just repeat the message several times. The hope is that the channel corrupts only a minority of these repetitions. This way the receiver will notice that a transmission error occurred since the received data stream is not the repetition of a single message, and moreover, the receiver can recover the original message by looking at the received message in the data stream that occurs most often.
Because of the bad error correcting performance coupled with the low code rate (ratio between useful information symbols and actual transmitted symbols), other error correction codes are preferred in most cases. The chief attraction of the repetition code is the ease of implementation.
In the case of a binary repetition code, there exist two code words - all ones and all zeros - which have a length of . Therefore, the minimum Hamming distance of the code equals its length . This gives the repetition code an error correcting capacity of (i.e. it will correct up to errors in any code word).
If the length of a binary repetition code is odd, then it's a perfect code. The binary repetition code of length n is equivalent to the (n,1)-Hamming code.
Consider a binary repetition code of length 3. The user wants to transmit the information bits 101. Then the encoding maps each bit either to the all ones or all zeros code word, so we get the 111 000 111, which will be transmitted.
Let's say three errors corrupt the transmitted bits and the received sequence is 111 010 100. Decoding is usually done by a simple majority decision for each code word. That lead us to 100 as the decoded information bits, because in the first and second code word occurred less than two errors, so the majority of the bits are correct. But in the third code word two bits are corrupted, which results in an erroneous information bit, since two errors lie above the error correcting capacity.
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