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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.)
Examples of the latter include redundancy in language structure or statistical properties relating to the occurrence frequencies of letter or word pairs, triplets etc. The minimum channel capacity can be realized in theory by using the typical set or in practice using Huffman , Lempel–Ziv or arithmetic coding .
In the field of data compression, Shannon coding, named after its creator, Claude Shannon, is a lossless data compression technique for constructing a prefix code based on a set of symbols and their probabilities (estimated or measured).
The converse of the capacity theorem essentially states that () is the best rate one can achieve over a binary symmetric channel. Formally the theorem states: Formally the theorem states:
In information theory, the source coding theorem (Shannon 1948) [2] informally states that (MacKay 2003, pg. 81, [3] Cover 2006, Chapter 5 [4]): N i.i.d. random variables each with entropy H(X) can be compressed into more than N H(X) bits with negligible risk of information loss, as N → ∞; but conversely, if they are compressed into fewer than N H(X) bits it is virtually certain that ...
Given a prime number q and prime power q m with positive integers m and d such that d ≤ q m − 1, a primitive narrow-sense BCH code over the finite field (or Galois field) GF(q) with code length n = q m − 1 and distance at least d is constructed by the following method.
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.
There are many names for interaction information, including amount of information, [1] information correlation, [2] co-information, [3] and simply mutual information. [4] Interaction information expresses the amount of information (redundancy or synergy) bound up in a set of variables, beyond that which is present in any subset of those variables.