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 $n_M$ and a decoding error $\epsilon > 0$ with $\epsilon\in\mathbb{Q}$, the following holds: $R_{n_M}(\epsilon)>C(P,N)-\frac{1}{2^M}$. It is shown that there is a polynomial-time computable $N_*$ such that for sufficiently large $P_*\in\mathbb{Q}$, the sequences $\{R_{n_M}(\epsilon)\}_{{n_M}\in\mathbb{N}}$, where each $R_{n_M}(\epsilon)$ 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 $\{R_{n_M}(\epsilon)\}_{n_M\in\mathbb{N}}$ grows faster than any polynomial as $M$ increases. Consequently, it is shown that either the sequence of achievable rates $\{R_{n_M}(\epsilon)\}_{n_M\in\mathbb{N}}$ as a function of the blocklength, or the sequence of blocklengths $\{n_M\}_{M\in\mathbb{N}}$ corresponding to the achievable rates, is not a polynomial-time computable sequence.

翻译：自Polyanskiy、Poor和Verdú关于离散无记忆信道容量可达码有限码长性能的开创性工作以来，已有许多研究尝试为更具实际意义的信道寻求进一步的结果。然而，计算容量可达码的复杂度问题似乎至今尚未得到深入探究。本文针对最简单且非平凡的高斯信道——加性有色高斯噪声信道——研究了该问题。为评估计算复杂度，我们考察了复杂度类 $\mathrm{FP}_1$ 与 $\#\mathrm{P}_1$。$\mathrm{FP}_1$ 包含可由确定性图灵机在多项式时间内计算的函数，而 $\#\mathrm{P}_1$ 涵盖那些对可在多项式时间内验证的解进行计数的函数。学界普遍假设 $\mathrm{FP}_1\neq\#\mathrm{P}_1$。我们关注的问题是：对于给定的 $M \in \mathbb{N}$（其中 $M$ 表示 $C(P,N)$ 偏离程度的精度），在特定码长 $n_M$ 和解码错误率 $\epsilon > 0$（$\epsilon\in\mathbb{Q}$）条件下，确定何时满足 $R_{n_M}(\epsilon)>C(P,N)-\frac{1}{2^M}$。研究证明，存在一个多项式时间可计算的 $N_*$，使得对于足够大的 $P_*\in\mathbb{Q}$，若 $\mathrm{FP}_1\neq\#\mathrm{P}_1$，则满足前述条件的序列 $\{R_{n_M}(\epsilon)\}_{{n_M}\in\mathbb{N}}$ 无法在多项式时间内计算。因此，随着 $M$ 增大，计算序列 $\{R_{n_M}(\epsilon)\}_{n_M\in\mathbb{N}}$ 的复杂度增长速度将超越任何多项式。最终结果表明：要么作为码长函数的可达速率序列 $\{R_{n_M}(\epsilon)\}_{n_M\in\mathbb{N}}$，要么对应于可达速率的码长序列 $\{n_M\}_{M\in\mathbb{N}}$，两者中至少有一个不是多项式时间可计算序列。