Kraftdistribution

Kraftdistribution is not a widely standardized term in information theory, but it is sometimes used informally to describe a probability distribution that is derived from the codeword lengths of a prefix code and that respects Kraft’s inequality. In this sense, a kraftdistribution is the distribution over codewords implied by a given length function l(w) for a prefix code.

If a binary prefix code has codeword lengths l(w) for each word w in its codebook, Kraft’s inequality states that the sum over all codewords of 2^{-l(w)} is at most 1. When the sum equals 1, the natural kraftdistribution assigns probability p(w) = 2^{-l(w)} to each codeword w. If the sum is less than 1, a normalized distribution p(w) = 2^{-l(w)}/Z can be defined, where Z = sum_w 2^{-l(w)}.

This distribution is not unique: different prefix codes (with different sets of lengths) induce different kraftdistributions. It is primarily a theoretical construct for analyzing the relationship between codeword lengths and probabilities, and it can aid in understanding code efficiency and redundancy in source coding. The concept also connects to the idea that shorter codewords should carry higher probability mass in optimal coding schemes.

In practice, one might refer to “the distribution induced by a prefix code” rather than using the term kraftdistribution, to describe the p(w) proportional to 2^{-l(w)} that arises from a given code’s length profile. See also Kraft inequality, prefix codes, and optimal source coding.