Abstract: We provide a theoretical analysis of the optimal capacity of modern Hopfield models [Ramsauer et al., 2020] and $\textbf{K}$ernelized $\textbf{H}$opfield $\textbf{M}$odels ($\textbf{KHM}$s) [Wu et al., 2024a]. The key ingredient in this paper is establishing a connection between memory distribution and spherical codes, viewing the stored memory set as a type of spherical code.With this connection, we transform the memory storage problem with KHMs into a point arrangement problem.We demonstrate that the optimal capacity of kernelized Hopfield models is achieved by a feature map such that memories form an optimal spherical code in its feature space. Our main theoretical contributions are as follows:(i) An analysis of how KHMs achieve optimal memory capacity and the necessary conditions for achieving it.(ii) A sub-linear time algorithm for reaching optimal capacity with KHMs.(iii) An analysis on the scaling behavior of the required feature dimension relative to the number of memory patterns.Our experimental results validate these theoretical findings.