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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.07773 |
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| _version_ | 1866911953533796352 |
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| author | Boche, Holger Grigorescu, Andrea Schaefer, Rafael F. Poor, H. Vincent |
| author_facet | Boche, Holger Grigorescu, Andrea Schaefer, Rafael F. Poor, H. Vincent |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07773 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite Blocklength Performance of Capacity-achieving Codes in the Light of Complexity Theory Boche, Holger Grigorescu, Andrea Schaefer, Rafael F. Poor, H. Vincent Information Theory 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. |
| title | Finite Blocklength Performance of Capacity-achieving Codes in the Light of Complexity Theory |
| topic | Information Theory |
| url | https://arxiv.org/abs/2407.07773 |