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A sphere packing bound (SPB) with a prefactor that is polynomial in the block length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is established for codes on a length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> product channel <inline-formula> <tex-math notation="LaTeX">$W_{[1,n]}$ </tex-math></inline-formula>, assuming that the maximum order 1/2 R&#x00E9;nyi capacity among the component channels, i.e. <inline-formula> <tex-math notation="LaTeX">$\max _{t\in [1,n]} C_{ {\scriptstyle {}^{\scriptstyle 1}}\hspace {-0.224em}/\hspace {-0.112em}{\scriptstyle 2},W_{t}}$ </tex-math></inline-formula>, is <inline-formula> <tex-math notation="LaTeX">$\mathit {O}(\ln n)$ </tex-math></inline-formula>. The reliability function of the discrete stationary product channels with feedback is bounded from above by the sphere packing exponent. Both results are proved by first establishing a non-asymptotic SPB. The latter result continues to hold under a milder stationarity hypothesis.

The Sphere Packing Bound via Augustin’s Method