Identification via Binary Uniform Permutation Channel

We study message identification over the binary uniform permutation channels. For DMCs, the number of identifiable messages grows doubly exponentially. Identification capacity, the maximum second-order exponent, is known to be the same as the Shannon capacity of a DMC. We consider a binary uniform permutation channel where the transmitted vector is permuted by a permutation chosen uniformly at random. Permutation channels support reliable communication of only polynomially many messages. While this implies a zero second-order identification rate, we prove a “soft” converse result showing that even non-zero first-order identification rates are not achievable with a power-law decay of error probability for identification over binary uniform permutation channels. To prove the converse, we use a sequence of steps to construct a new identification code with a simpler structure and then use a lower bound on the normalized maximum pairwise intersection of a set system on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\{0, \ldots, n\}$</tex>.