Refinement of the Sphere-Packing Bound: Asymmetric Channels

We provide a refinement of the sphere-packing bound for constant composition codes over asymmetric discrete memoryless channels that improves the pre-factor in front of the exponential term. The order of our pre-factor is $\mathrm{\Omega }(N^{-1\mathrm{/}2(1+ϵ+{\rho }_R^{\ast })})$ for any $ϵ>0$, where ${\rho }_R^{\ast }$ is the maximum absolute-value subdifferential of the sphere-packing exponent at rate $R$ and $N$ is the blocklength.