Finite Blocklength Performance of Capacity-achieving Codes in the Light of Complexity Theory

Since the work of Polyanskiy, Poor and Verd\'u on the finite blocklength performance of capacity-achieving codes for discrete memoryless channels, many papers have attempted to find further results for more practically relevant channels. However, it seems that the complexity of computing capacity-achieving codes has not been investigated until now. We study this question for the simplest non-trivial Gaussian channels, i.e., the additive colored Gaussian noise channel. To assess the computational complexity, we consider the classes $\mathrm{FP}1$ and $#\mathrm{P}1$. $\mathrm{FP}1$ includes functions computable by a deterministic Turing machine in polynomial time, whereas $#\mathrm{P}1$ encompasses functions that count the number of solutions verifiable in polynomial time. It is widely assumed that $\mathrm{FP}1\neq#\mathrm{P}1$. It is of interest to determine the conditions under which, for a given $M \in \mathbb{N}$, where $M$ describes the precision of the deviation of $C(P,N)$, for a certain blocklength $nM$ and a decoding error $\epsilon > 0$ with $\epsilon\in\mathbb{Q}$, the following holds: $R{nM}(\epsilon)>C(P,N)-\frac{1}{2^M}$. It is shown that there is a polynomial-time computable $N$ such that for sufficiently large $P_\in\mathbb{Q}$, the sequences ${R{nM}(\epsilon)}{{nM}\in\mathbb{N}}$, where each $R{nM}(\epsilon)$ satisfies the previous condition, cannot be computed in polynomial time if $\mathrm{FP}1\neq#\mathrm{P}1$. Hence, the complexity of computing the sequence ${R{nM}(\epsilon)}{nM\in\mathbb{N}}$ grows faster than any polynomial as $M$ increases. Consequently, it is shown that either the sequence of achievable rates ${R{nM}(\epsilon)}{nM\in\mathbb{N}}$ as a function of the blocklength, or the sequence of blocklengths ${nM}{M\in\mathbb{N}}$ corresponding to the achievable rates, is not a polynomial-time computable sequence.