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 $1/2$ Renyi capacity among the component channels, i.e. $\max_{ t\in[1,n]} C_{ 1/2,W_{ t}}$, 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.
Categories: cs.IT math.IT