2018 Fiscal Year Final Research Report
Optimal codes derived from multifold factorizations of cyclic groups and a certain kind of difference sets
| Project/Area Number |
16K05269
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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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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | Gifu University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
神保 雅一 中部大学, 現代教育学部, 教授 (50103049)
宮本 暢子 東京理科大学, 理工学部情報科学科, 教授 (20318207)
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| Research Collaborator |
FU Hung-Lin
FU Chin-Mei
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | multifold factorization / cyclic group / line spread / conflict-avoiding code / DSS |
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| Outline of Final Research Achievements |
For a given factor which is a multiset over a cyclic group, sufficient conditions for its complement factor to give the minimum factorization of the cyclic group have been given. By applying the result on the case where a multifold factorization has a periodic factor, the maximum number of multifold spreads partitioning the lines in PG(2n-1,q) has been determined for a positive integer n and q=3,4. For nonnegative integers u and e, and a prime p≡1,3 (mod 8), the upper bound of the number of codewords for a conflict-avoiding code of length pow(3,3u+1)*pow(p,e) and weight 3 has been also derived together with the condition for attaining the bound. Besides, for m=7,8,9,10 and a positive integer t, some sets of parameters for difference system of sets (DSS) over a finite field of prime order p=2mt+1 such that the blocks are cyclotomic cosets of index 2t have been obtained.
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| Free Research Field |
デザイン理論
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| Academic Significance and Societal Importance of the Research Achievements |
巡回群の分解問題や差集合の存在問題は，デザイン理論やグラフ理論，整数論等の基礎理論だけでなく，符号や暗号等，工学的にも広く応用できることが知られている．本研究で得られた有限巡回群の多重分解の結果は，より良い性質をもつ量子ジャンプ符号や秘密分散法の導出に貢献する基礎理論となる．また，衝突回避符号，および畳み込み符号に関連するDifference System of Sets(DSS)の結果により，これまで未解決であったパラメータをもつ多元接続通信のための最適なプロトコル系列の存在を明らかにできた．
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