We study message identification over a $q$-ary uniform permutation channel, where the transmitted vector is permuted by a permutation chosen uniformly at random. For discrete memoryless channels (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 the DMC. Permutation channels support reliable communication of only polynomially many messages. A simple achievability result shows that message sizes growing as $2^{c_nn^{q-1}}$ are identifiable for any $c_n\rightarrow 0$. We prove two converse results. A ``soft'' converse shows that for any $R>0$, there is no sequence of identification codes with message size growing as $2^{Rn^{q-1}}$ with a power-law decay ($n^{-\mu}$) of the error probability. We also prove a ``strong" converse showing that for any sequence of identification codes with message size $2^{Rn^{q-1}\log n}$ ($R>0$), the sum of type I and type II error probabilities approaches at least $1$ as $n\rightarrow \infty$. To prove the soft converse, we use a sequence of steps to construct a new identification code with a simpler structure which relates to a set system, and then use a lower bound on the normalized maximum pairwise intersection of a set system. To prove the strong converse, we use results on approximation of distributions.

翻译：我们研究在$q$元均匀置换信道上的消息识别问题，其中传输向量通过一个均匀随机选择的置换进行重排。对于离散无记忆信道（DMCs），可识别消息的数量呈双指数增长。已知识别容量（即最大二阶指数）与DMC的香农容量相同。置换信道仅支持多项式数量消息的可靠传输。一个简单的可达性结果表明，对于任意趋于零的$c_n$，消息规模以$2^{c_nn^{q-1}}$增长时是可识别的。我们证明了两类逆命题。一个“弱”逆命题表明：对于任意$R>0$，不存在消息规模以$2^{Rn^{q-1}}$增长且错误概率具有幂律衰减（$n^{-\mu}$）的识别码序列。我们还证明了一个“强”逆命题：对于任意消息规模为$2^{Rn^{q-1}\log n}$（$R>0$）的识别码序列，当$n\rightarrow \infty$时，第一类与第二类错误概率之和至少趋近于$1$。为证明弱逆命题，我们通过一系列步骤构造了一个具有更简单结构的新识别码，该结构与集合系相关联，随后利用集合系归一化最大两两交集的下界。为证明强逆命题，我们使用了分布逼近的相关结果。