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

Holger Boche, Andrea Grigorescu, Rafael F. Schaefer, H. Vincent Poor

TL;DR

It is shown that the sequence of achievable rates as a function of the blocklength, or the sequence of blocklengths corresponding to the achievable rates, is not a polynomial-time computable sequence.

Abstract

Since the work of Polyanskiy, Poor and Verdú 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 $ε> 0$ with $ε\in\mathbb{Q}$, the following holds: $R_{n_M}(ε)>C(P,N)-\frac{1}{2^M}$. It is shown that there is a polynomial-time computable $N_*$ such that for sufficiently large $P_*\in\mathbb{Q}$, the sequences $\{R_{n_M}(ε)\}_{{n_M}\in\mathbb{N}}$, where each $R_{n_M}(ε)$ 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 $\{R_{n_M}(ε)\}_{n_M\in\mathbb{N}}$ grows faster than any polynomial as $M$ increases. Consequently, it is shown that either the sequence of achievable rates $\{R_{n_M}(ε)\}_{n_M\in\mathbb{N}}$ 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.

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

TL;DR

It is shown that the sequence of achievable rates as a function of the blocklength, or the sequence of blocklengths corresponding to the achievable rates, is not a polynomial-time computable sequence.

Abstract

Since the work of Polyanskiy, Poor and Verdú 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 and . includes functions computable by a deterministic Turing machine in polynomial time, whereas encompasses functions that count the number of solutions verifiable in polynomial time. It is widely assumed that . It is of interest to determine the conditions under which, for a given , where describes the precision of the deviation of , for a certain blocklength and a decoding error with , the following holds: . It is shown that there is a polynomial-time computable such that for sufficiently large , the sequences , where each satisfies the previous condition, cannot be computed in polynomial time if . Hence, the complexity of computing the sequence grows faster than any polynomial as increases. Consequently, it is shown that either the sequence of achievable rates as a function of the blocklength, or the sequence of blocklengths corresponding to the achievable rates, is not a polynomial-time computable sequence.
Paper Structure (1 theorem, 4 equations, 1 figure)

This paper contains 1 theorem, 4 equations, 1 figure.

Key Result

Theorem 1

Let $B$ be a polynomial time computable number representing the bandwidth. There exists a strictly positive and polynomial time computable noise power spectrum $N_*$ such that for all sufficient large rational power constraint $P_*$ and for all rational $\epsilon > 0$, the computation of achievable is in $\#\mathrm{P}_1$. If $\mathrm{FP}_1 \neq \#\mathrm{P}_1$, then for $N_*$ and for every suffic

Figures (1)

  • Figure 1: The red line represents the band-limited ACGN capacity $C=C(P,N_*)$ for the power spectral density. $N_*$ and the power constraint $P$ in the asymptotic regime. The black curve represents the finite blocklength achievable rate $R_{n}(\epsilon)$ for some fixed $\epsilon>0$. For $n_1$ we have $R_{n_1}(\epsilon)>C-\frac{1}{2}$ and for $n_2$ we have $R_{n_2}(\epsilon)>C-\frac{1}{4}$. Fig. from boche2024characterizationj.

Theorems & Definitions (11)

  • Definition 1: Class $\mathrm{FP}$
  • Definition 2: Classes $\mathrm{FP}_1$ and $\#\mathrm{P}_1$
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 1 more