Background On Error-correcting Codes
An original message and an encoded version are both composed in an alphabet of q letters. Each code word contains n letters. The original message (of length m) is shorter than n letters. The message is converted into an n-letter codeword by an encoding algorithm, transmitted over a noisy channel, and finally decoded by the receiver. The decoding process interprets a garbled codeword, referred to as simply a word, as the valid codeword "nearest" the n-letter received string.
Mathematically, there are exactly qm possible messages of length m, and each such message can be regarded as a vector of length m. The encoding scheme converts an m-dimensional vector into an n-dimensional vector. Exactly qm valid codewords are possible, but any one of qn garbled codewords (words) can be received, because the noisy channel might distort one or more of the n letters while the codeword is being transmitted.
Read more about this topic: Hamming Bound
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