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Let each source symbol from the alphabet = {,, …,} be encoded into a uniquely decodable code over an alphabet of size with codeword lengths ,, …,. Then = Conversely, for a given set of natural numbers ,, …, satisfying the above inequality, there exists a uniquely decodable code over an alphabet of size with those codeword lengths.
A code is uniquely decodable if its extension is § non-singular.Whether a given code is uniquely decodable can be decided with the Sardinas–Patterson algorithm.. The mapping = {,,} is uniquely decodable (this can be demonstrated by looking at the follow-set after each target bit string in the map, because each bitstring is terminated as soon as we see a 0 bit which cannot follow any ...
For example, a code with code {9, 55} has the prefix property; a code consisting of {9, 5, 59, 55} does not, because "5" is a prefix of "59" and also of "55". A prefix code is a uniquely decodable code: given a complete and accurate sequence, a receiver can identify each word without requiring a special marker between words. However, there are ...
Kraft's inequality in some cases provides a quick way to exclude the possibility that a given code is uniquely decodable. Prefix codes and block codes are important classes of codes which are uniquely decodable by definition. Timeline of information theory; Post's correspondence problem is similar, yet undecidable.
Muller codes require that n be a power of 2 and that the Hadamard matrix be equivalent to the matrix constructed by Sylvester's method. The Hadamard code is a locally decodable code, which provides a way to recover parts of the original message with high probability, while only looking at a small fraction of the received word.
The prefix code {00, 11} is not self-synchronizing; while 0, 1, 01 and 10 are not codes, 00 and 11 are. The prefix code {ab,ba} is not self-synchronizing because abab contains ba. The prefix code b ∗ a (using the Kleene star) is not self-synchronizing (even though any new code word simply starts after a) because code word ba contains code word a.
In coding theory, an expander code is a [,] linear block code whose parity check matrix is the adjacency matrix of a bipartite expander graph.These codes have good relative distance (), where and are properties of the expander graph as defined later, rate (), and decodability (algorithms of running time () exist).
Fig 1 is an example of a SCCC. Fig. 1. SCCC Encoder. The example encoder is composed of a 16-state outer convolutional code and a 2-state inner convolutional code linked by an interleaver. The natural code rate of the configuration shown is 1/4, however, the inner and/or outer codes may be punctured to achieve higher code rates as needed.