L-identification for uniformly distributed sources and the q-ary identification entropy of second order

• Authors: Christian Heup
• Venue: Information Theory, Combinatorics, and Search Theory
• Year: 2013

In this article we generalize the concept of identification for sources, which was introduced by Ahlswede, to the concept of L-identification for sources. This means that we do not only consider a discrete source but a discrete memoryless source (DMS) with L outputs. The task of L-identification is now to check for any previously given output whether it is part of the L outputs of the DMS. We establish a counting lemma and use it to show that, if the source is uniformly distributed, the L-identification symmetric running time asymptotically equals the rational number $$K_{L, q}=-\sum_{l=1}^L(-1)^l{L \choose l}\frac{q^l}{q^l-1}\enspace.$$ We then turn to general distributions and aim to establish a lower bound for the symmetric 2-identification running time. In order to use the above asymptotic result we first concatenate a given code sufficiently many times and show that for 2-identification the uniform distribution is optimal, thus yielding a first lower bound. This lower bound contains the symmetric (1-)identification running time negatively signed so that (1-)identification entropy cannot be applied immediately. However, using the fact that the (1-)identification entropy is attained iff the probability distribution consists only of q-powers, we can show that our lower bound is in this case also exactly met for 2-identification. We then prove that the obtained expression is in general a lower bound for the symmetric 2-identification running time and that it obeys fundamental properties of entropy functions. Hence, the following expression is called the q-ary identification entropy of second order$$H_{\mbox{\tiny ID}}^{2,q}(P) =2\frac{q}{q-1}\left(1-\sum_{u\in{\mathcal U}}p_u^2\right)-\frac{q^2}{q^2-1}\left(1-\sum_{u\in{\mathcal U}}p_u^3\right)\enspace.$$