Generalizing the Kraft-McMillan Inequality to restricted languages

초록

Let l1, l2,...,)ln be a (possibly infinite) sequence of nonnegative integers and Σ some D-ary alphabet. The Kraft-Inequality states that l1, l2,...,ln are the lengths of the words in some prefix (free) code over E if and only if Σi=1 nD-4 ≤ 1. Furthermore, the code is exhaustive if and only if equality holds. The McMillan Inequality states that if l1, l2,..., ln are the lengths of the words in some uniquely decipherable code, then the same condition holds. In this note we examine how the Kraft-McMillan Inequality conditions for the existence of a prefix or uniquely decipherable code change when the code is not only required to be prefix but all of the codewords are restricted to belong to a given specific language ℒ. For example, ℒ might be all words that end in a particular pattern or, if Σ is binary, might be all words in which the number of zeros equals the number of ones. © 2005 IEEE.