The Sphere Packing Bound via Augustin’s Method

A sphere packing bound (SPB) with a prefactor that is polynomial in the block length $n$ is established for codes on a length $n$ product channel $W_{[1,n]}$ , assuming that the maximum order 1/2 Rényi capacity among the component channels, i.e. $\max _{t\in [1,n]} C_{ {\scriptstyle {}^{\scriptstyle 1}}\hspace {-0.224em}/\hspace {-0.112em}{\scriptstyle 2},W_{t}}$ , is $\mathit {O}(\ln n)$ . 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.