2022 Fiscal Year Final Research Report
Applications of probabilistic combinatorics and extremal set theory to deriving bounds in classical and quantum coding theory
| Project/Area Number |
20K11668
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 60010:Theory of informatics-related
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| Research Institution | Chiba University |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2023-03-31
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| Keywords | 符号理論 / 組合せ論 / 極値集合論 / 確率論 / 自己同期符号 / 組合せ符号 / 誤り訂正符号 |
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| Outline of Final Research Achievements |
The primary purpose of this research project has been to find and explore novel applications of probabilistic and extremal combinatorics to coding theory and help develop a theory that connects the two types of combinatorics and coding theory in new ways. For this purpose, we investigated several well-known unsolved coding-theoretic problems that had resisted successful applications of probabilistic arguments and related extremal set-theoretic approaches. The highlight of the results obtained by this research project is the resolution of an important problem in coding theory that had been open for more than 50 years. We proved that the asymptotic rate of what is known as an optimal difference system of sets achieves the well-known Levenshtein bound. This means that, in theory, we can develop a computationally efficient synchronization system even under the presence of strong additive noise.
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| Free Research Field |
符号理論
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| Academic Significance and Societal Importance of the Research Achievements |
本研究で得られた成果はさまざまであるが，その最大のものは前項で述べた，最適 DSS の漸近符号化率が Levenshtein 限界を如何なる要求雑音耐性水準においても達成することを証明したことである．最適 DSS はデジタル通信における送信者と受信者の同期を，雑音下においても高い信頼性を保証しつつ効率的に行うための数学的構造物である．本研究では DSS の理論限界を明らかにするとともに，簡単に漸近最適である DSS を構成するアルゴリズムを提示しており，数学的にも，情報理論的にも，また電気電子工学的にも興味深い成果である．
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