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 Ω(N
^{-0.5(1+ε+ρ
R
*}
)) for any ϵ > 0, where ρ
_R
* is the left derivative of the sphere-packing exponent at rate R and N is the blocklength.