In error detection and correction

Error detection and correction

In information theory and coding theory with applications in computer science and telecommunication, error detection and correction or error control are techniques that enable reliable delivery of digital data over unreliable communication channels...

, majority logic decoding is a method to decode repetition code

Repetition code

In coding theory, the repetition code is one of the most basic 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...

s, based on the assumption that the largest number of occurrences of a symbol was the transmitted symbol.

Theory

In a binary alphabet made of , if a repetition code is used, then each input bit is mapped to the code word

Code word

In communication, a code word is an element of a standardized code or protocol. Each code word is assembled in accordance with the specific rules of the code and assigned a unique meaning...

 as a string of -replicated input bits. Generally , an odd number.

The repetition codes can detect up to transmission errors. Decoding errors occur when the more than these transmission errors occur. Thus, assuming bit-transmission errors are independent, the probability of error for a repetition code is given by , where is the error over the transmission channel.

Assumptions

The code word is , where , an odd number.

• Calculate the Hamming weight
  Hamming weight
  The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string of bits, this is the number of 1's in the string...
  
  of the repetition code.
• if , decode code word to be all 0's
• if , decode code word to be all 1's

Example

In a code, if R=[1 0 1 1 0], then
it would be decoded as,

• , , so R'=[1 1 1 1 1]
• Hence the transmitted message bit was 1.