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 $1\mathrm{/}2$ Renyi capacity among the component channels, i.e. ${\max }_{t\in [1,n]}{C}_{1\mathrm{/}2,W_t}$, is $O(\ln n)$. 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.