An inequality on guessing and its application to sequential decoding

Let (X, Y) be a pair of discrete random variables with X taking one of M possible values. Suppose the value of X is to be determined, given the value of Y, by asking quest ions of the form “Is X equal to z?” until the answer is “Yes.” Let G(r 1 y) denote the number of guesses in any such guessing scheme when X = x, Y = y. W e prove that 1 1+/J E[G(X IY)‘] 2 (l+InM)-P~ for any p > 0. This provides an operational characterization of RCnyi’s entropy. Next we apply this inequality to the estimation of the computat ional complexity of sequential decoding. For this, we regard X as the input, Y as the output of a communicat ion channel. Given Y, the sequential decoding algorithm works essentially by guessing X, one value at a time, until the guess is correct. Thus the computat ional complexity of sequential decoding, which is a random variable, is given by a guessing function G(X 1 ‘Y) that is def ined by the order in which nodes in the tree code are hypothesized by the decoder. This observation, combined with the above lower bound on moments of G(X 1 Y), yields lower bounds on moments of computat ion in sequential decoding. The present approach enables the determination of the (previously known) cutoff rate of sequential decoding in a simple manner; it also yields the (previously unknown) cutoff rate region of sequential decoding for mult iaccess channels. These results hold for memoryless channels with finite input alphabets. Zndex Terms-Guessing, Holder’s inequality, sequential decoding, RCnyi’s entropy.