Enumerative Source

73 The limiting value of the lower bound for iI in c+c.ois (19) as H(R)-/21(E) = 3 log, cP/&. instead of two variables. Jelinek [8] has shown that the Lagrange multiplier method [4], [S] of evaluating the rate-distortion function for balanced distortion measures can be extended to nonbalanced measures. In a sense, the Chernoff bounding method of Lemmas 1 and 2 takes the place of the Kuhn-Tucker conditions in the Lagrange multiplier method. However, the assumption of row balance plays an important role in the proof of Lemma 1. It is not clear whether this assumption can be relaxed in the same way that the corresponding assumption of column balance was relaxed by Jelinek [S]. As in the previous example, it is possible to obtain the bound of Lemma 4 by an alternative method. In this case, the set G(t,r) is just a sphere in RN with radius (Nt) " '. Hence y[G(t,x)] is the volume of this sphere, p[G(t,x)] = s. An application of Stirling's formula to this expression gives an expression that agrees with the bound in (16) (up to factors that do not increase exponentially with N). VII. CONCLUDING REMARKS For certain classes of distortion measures we have succeeded in establishing generalizations of the Kraft inequality. As is the case with the Kraft inequality for unique decipherability, the inequality is independent of any particular probability distribution on the source and depends only on the fidelity of reproduction according to our definition of .s-decodability. This separation of fidelity from source probability may sometimes be useful. It does not seem possible to get a simple extension of the Kraft inequality for all distortion measures. Some assumption , like row balance in the discrete case or a difference measure in the continuous case, seems necessary in order to permit treatment of d(cQ) as a function of one variable It is conceivable that tighter versions of the Kraft inequality exist in those cases where our lower bound on mean length does not coincide with the rate-distortion function. In any case, the versions presented here have the virtue of simplicity. A measure of asymptotic efficiency for tests of a hypothesis based on a sum of observations, " Ann. Abshact-Let S be a given subset of binary n-sequences. We provide an explicit scheme for calculating the index of any sequence in S according to its position in the lexicographic ordering of S. A …