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 subexponential factor in front of the exponent. The order of our subexponential factor is $\Omega (N^{-0.5(1+\epsilon+\rho_{R}^{\ast})})$ for any $\epsilon>0$, where $\rho_{R}^{\ast}$ is the left derivative of the sphere-packing exponent at rate $R$ and $N$ is the blocklength.