The sphere packing bound for memoryless channels

Sphere packing bounds (SPBs)—with prefactors that are polynomial in the block length—are derived for codes on two families of memoryless channels using Augustin’s method: (possibly nonstationary) memoryless channels with (possibly multiple) additive cost constraints and stationary memoryless channels with convex constraints on the composition (i.e., empirical distribution, type) of the input codewords. A variant of Gallager’s bound is derived in order to show that these sphere packing bounds are tight in terms of the exponential decay rate of the error probability with the block length under mild hypotheses.