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[4] [5] The redundancy allows the receiver not only to detect errors that may occur anywhere in the message, but often to correct a limited number of errors. Therefore a reverse channel to request re-transmission may not be needed. The cost is a fixed, higher forward channel bandwidth.
The quantity is called the relative redundancy and gives the maximum possible data compression ratio, when expressed as the percentage by which a file size can be decreased. (When expressed as a ratio of original file size to compressed file size, the quantity R : r {\displaystyle R:r} gives the maximum compression ratio that can be achieved.)
Shannon–Fano–Elias coding produces a binary prefix code, allowing for direct decoding.. Let bcode(x) be the rational number formed by adding a decimal point before a binary code.
For example, 1011 is encoded (using the non-systematic form of G at the start of this section) into 01 1 0 011 0 where blue digits are data; red digits are parity bits from the [7,4] Hamming code; and the green digit is the parity bit added by the [8,4] code. The green digit makes the parity of the [7,4] codewords even.
A cyclic redundancy check (CRC) is a non-secure hash function designed to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors.
The method was the first of its type, the technique was used to prove Shannon's noiseless coding theorem in his 1948 article "A Mathematical Theory of Communication", [1] and is therefore a centerpiece of the information age.
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Here, () = = is the entropy, and Shannon's source coding theorem says that any code must have an average length of at least (). Hence we see that the Shannon–Fano code is always within one bit of the optimal expected word length.