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
Famous quotes containing the words background and/or codes:
“They were more than hostile. In the first place, I was a south Georgian and I was looked upon as a fiscal conservative, and the Atlanta newspapers quite erroneously, because they didn’t know anything about me or my background here in Plains, decided that I was also a racial conservative.”
—Jimmy Carter (James Earl Carter, Jr.)
“... until both employers’ and workers’ groups assume responsibility for chastising their own recalcitrant children, they can vainly bay the moon about “ignorant” and “unfair” public criticism. Moreover, their failure to impose voluntarily upon their own groups codes of decency and honor will result in more and more necessity for government control.”
—Mary Barnett Gilson (1877–?)