Research on combinatorics over Galois rings

• Project/Area Number: 20540014
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kanazawa University
• Principal Investigator: YAMADA Mieko 金沢大学, 数物科学系, 教授 (70130226)
• Co-Investigator(Kenkyū-buntansha): 菅野 孝史 金沢大学, 数物科学系, 教授 (30183841)
• Project Period (FY): 2008 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2008: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: ガロア環 / 差集合 / 符号理論 / ガウス和 / designs / Hadamard行列 / ガロア和

Research Abstract

We constructed infinite families of difference sets over Galois rings of characteristic an even power of 2. The difference set over a Galois ring of characteristic 2^n is embedded in the ideal part of the difference set over a Galois ring of characteristic 2^n+2 . We introduced a new operation in a Galois ring and the Gauss sums associated with the character under this new operation play an important role of the proof. Furthermore, we proved there exist Reed-Muller codes with embedding system and showed several properties of them.

Research Products

• [Journal Article] Divisible difference family from Galois rings GR(4,n) and Hadamard matrices (2013)
  • Author(s): Koji Momihara and Mieko Yamada
  • Journal Title: Designs, Codes andCryptography Volume: 掲載決定
  • URL: http://link.springer.com/journal/10623
  • Peer Reviewed
• [Journal Article] Difference sets overGalois rings with odd extension degree and characteristic an even power of 2 (2013)
  • Author(s): Mieko Yamada
  • Journal Title: Designs, Codes and Cryptography Volume: 67 Issue: 1 Pages: 37-57
  • DOI: 10.1007/s10623-011-9584-z
  • Peer Reviewed
• [Journal Article] Difference sets over Galois rings with odd extension degree and characteristic an even power of 2 (2013)
  • Author(s): Mieko Yamada
  • Journal Title: Designs, Codes and Cryptography Volume: 67 Pages: 37-57
  • Peer Reviewed
• [Journal Article] Reed-Muller codes over Galois rings of characteristic 2^n (2012)
  • Author(s): Soryo Kawasaki and Mieko Yamada
  • Journal Title: Science Reports of Kanazawa University Volume: 56 Pages: 1-13
  • NAID: 120005307433
  • URL: http://dspace.lib.kanazawa-u.ac.jp/dspace/bulletin/sci_rep
  • Peer Reviewed
• [Journal Article] Reed-Muller codes over Galois rings of characteristic 2^n (2012)
  • Author(s): Soryo Kawasaki and Mieko Yamada
  • Journal Title: Science Reports of Kanazawa Univesity Volume: 56 Pages: 1-13
  • NAID: 120005307433
  • Peer Reviewed
• [Journal Article] Generalized extended Hamming codes over Galois rings of characteristic 2^n (2011)
  • Author(s): Muhammad Ilyas and Mieko Yamada
  • Journal Title: Mathematical Science and Applications, International Symposium on Computational Science 2011 Volume: 34 Pages: 139-150
  • Peer Reviewed
• [Journal Article] Perfect difference systems of set and Jacobi sums (2009)
  • Author(s): R.Fuji-Hara, K.Momihara, Mieko Yamada
  • Journal Title: Discrete Mathematics Volume: 309 Issue: 12 Pages: 3954-3961
  • DOI: 10.1016/j.disc.2008.11.009
  • Peer Reviewed
• [Journal Article] Perfect difference systems of set and Jacobi sums (2009)
  • Author(s): R.Fuji-Hara, K.Momihara, Mieko Yamada,(論文名)
  • Journal Title: Discrete Mathematics 309 Pages: 3954-3961
  • Peer Reviewed
• [Journal Article] Perfect difference systems of sets and Jacobi sums (2009)
  • Author(s): R.Fuji-Hara, K.Momihara, Mieko Yamada
  • Journal Title: Discrete Mathematics 309 Pages: 3954-3961
  • Peer Reviewed
• [Journal Article] Perfect difference systems of set and Jacobi sums (2009)
  • Author(s): R. Fuji-Hara, K. Momihara, Mi eko Yamada
  • Journal Title: Discrete Mathematics 未定(掲載確定)
  • Peer Reviewed
• [Journal Article] Self-dual Z4-codes of Type IV generated by skew-Hadamard matrices and conference matrices (2008)
  • Author(s): Mieko Yamada
  • Journal Title: Australasian Journal of Combinatorics Volume: 42 Pages: 177-188
  • URL: http://ajc.maths.u9.edu.au
  • Peer Reviewed
• [Journal Article] Self-dual Z_4-codes of Type IV generated by skew-Hadamard matrices and conference matrices (2008)
  • Author(s): Mieko Yamada(論文名)
  • Journal Title: Australasian Journal of Combinatorics 42 Pages: 177-188
  • Peer Reviewed
• [Journal Article] Self-dual Z_4-codes of Type IV generated by skew-Hadamard matri ces and conference matrices (2008)
  • Author(s): Mieko Yamada
  • Journal Title: Australasian Journal of Combinatorics 42 Pages: 177-188
  • Peer Reviewed
• [Presentation] 標数4のガロア環上のKerdock code から得られるassociation scheme (2012)
  • Author(s): 河崎宗亮、山田美枝子
  • Organizer: 2012年応用数学合同研究集会
  • Place of Presentation: 龍谷大学(滋賀)
  • Year and Date: 2012-12-21
• [Presentation] Z_p2 上のPreparata code の復号アルゴリズム (2012)
  • Author(s): 皆川泰蔵、山田美枝子
  • Organizer: 2012年応用数学合同研究集会2012年応用数学合同研究集会
  • Place of Presentation: 龍谷大学(滋賀)
  • Year and Date: 2012-12-20
• [Presentation] The constructions of unrealButson-type Hadamard matricesBH(n,6)'s (2011)
  • Author(s): Pritta Etriana Putri, Mieko Yamada
  • Organizer: 2011年応用数学合同研究集会
  • Place of Presentation: 龍谷大学(滋賀)
  • Year and Date: 2011-12-16
• [Presentation] 標数2nのガロア環の拡張BCH符号の部分符号について (2011)
  • Author(s): 大江純矢、山田美枝子
  • Organizer: 2011年応用数学合同研究集会
  • Place of Presentation: 龍谷大学(滋賀)
  • Year and Date: 2011-12-15
• [Presentation] Difference sets,divisible families and codes over Galois rings of characteristic 2^n (2011)
  • Author(s): Mieko Yamada
  • Organizer: Workshop on Combinatorial Designs
  • Place of Presentation: National University of Singapore(シンガポール)
  • Year and Date: 2011-06-02
• [Presentation] Generalized extended Hamming codes over Galois rings of characteristic 2^n (2011)
  • Author(s): Muhammad Ilyas, Mieko Yamada
  • Organizer: International Symposium on Computational Science 2011
  • Place of Presentation: 金沢大学(石川)
  • Year and Date: 2011-02-16
• [Presentation] Generalized extended Hamming codes over Galois rings of characteristic 2^n (2011)
  • Author(s): Muhammad Ilyas, Mieko Yamada(発表標題)
  • Organizer: International Symposium on Computational Science 2011
  • Place of Presentation: 金沢大(石川県)
  • Year and Date: 2011-02-16
• [Presentation] Generalized extended Hamming codes over Galois rings of characteristic.2^n (2011)
  • Author(s): Muhammad Ilyas, Mieko Yamada
  • Organizer: International Symposium on Computational Science 2011
  • Place of Presentation: 金沢大(石川県)
  • Year and Date: 2011-02-16
• [Presentation] Difference sets and Reed-Muller codes over Galois rings of characteristic 2^n (2011)
  • Author(s): Mieko Yamada
  • Organizer: Linear Algebraic Techniques in Combinatorics/Graph Theory
  • Place of Presentation: BanffInternational Research Station(カナダ)
  • Year and Date: 2011-02-01
• [Presentation] Difference sets and Reed-Muller codes over Galois rings of characteristic 2^n (2011)
  • Author(s): Mieko Yamada(発表標題)
  • Organizer: Linear Algebraic Techniques in Combinatorics/Graph Theory
  • Place of Presentation: Banff International Research Station(Canada)
  • Year and Date: 2011-02-01
• [Presentation] Difference sets and Reed-Muller codes over Galois rings of characteristi C 2^n (2011)
  • Author(s): Mieko Yamada
  • Organizer: Linear Algebraic Techniques in Combinatorics/GrapHh Theory
  • Place of Presentation: Banff International Research Station(Canada)(招待講演)
  • Year and Date: 2011-02-01
• [Presentation] 標数2^nのガロア環上のKerdock code (2010)
  • Author(s): 河崎宗亮、山田美枝子
  • Organizer: 2010年度応用数学合同研究集会
  • Place of Presentation: 龍谷大学(滋賀県)
  • Year and Date: 2010-12-17
• [Presentation] Preparata oodes over Galois rings of chazacteriatic 2^n (2010)
  • Author(s): Muhammad Ilyas, 山田美枝子
  • Organizer: 2010年度応用数学合同研究集会
  • Place of Presentation: 龍谷大学(滋賀県)
  • Year and Date: 2010-12-17
• [Presentation] 標数2^<n>のガロア環上の Reed-Muller codes (2009)
  • Author(s): 河崎宗亮, 山田美枝子
  • Organizer: 2009年度応用数学合同研究集会
  • Place of Presentation: 龍谷大学 (滋賀県)
  • Year and Date: 2009-12-18
• [Presentation] Difference sets over Galois rings GR(2^n,s) of odd extensions (2009)
  • Author(s): Mieko Yamada
  • Organizer: British Combinatorial Conference 2009
  • Place of Presentation: University of St. Andrews(英国)
  • Year and Date: 2009-07-09
• [Presentation] Difference sets over Galois rings GR(2^n, s)of odd extensions (2009)
  • Author(s): Mieko Yamada(発表標題)
  • Organizer: British Combinatorial Conference 2009
  • Place of Presentation: University of St.Andrews(英国)
  • Year and Date: 2009-07-09
• [Presentation] Difference sets over Galois rings GR (2^<n>, s) of odd extensions (2009)
  • Author(s): 山田美枝子
  • Organizer: British Combinatorial Conference 2009
  • Place of Presentation: University of St. Andrews (英国)
  • Year and Date: 2009-07-09
• [Presentation] Difference sets over Galois rings (2009)
  • Author(s): 山田美枝子
  • Organizer: 夏の学校「Designs and Codes」
  • Place of Presentation: ヒルズサンピア山形 (山形県)
  • Year and Date: 2009-06-22
• [Presentation] 種々の差集合の構成とガウス和、ヤコビ和 (2008)
  • Author(s): Mieko Yamada
  • Organizer: 離散数学とその応用研究集会2008
  • Place of Presentation: 茨城大学(茨城)
  • Year and Date: 2008-08-21
• [Presentation] 種々の差集合の構成とガウス和、ヤコビ和 (2008)
  • Author(s): Mieko Yamada(発表標題)
  • Organizer: 離散数学とその応用研究集会2008
  • Place of Presentation: 茨城大学(水戸市)
  • Year and Date: 2008-08-21
• [Presentation] 種々の差集合の構成とガウス和、ヤコビ和 (2008)
  • Author(s): Mieko Yamada
  • Organizer: 離散数学とその応用研究集会2008
  • Place of Presentation: 水戸市
  • Year and Date: 2008-08-21
• [Presentation] Difference sets over Galois rings GR(2^n, s) for even n and even s (2008)
  • Author(s): Mieko Yamada
  • Organizer: Combinatorics 2008
  • Place of Presentation: Costermano(イタリア)
  • Year and Date: 2008-06-24
• [Presentation] Difference sets over Galois rings GR(2^t, s)for evenn and even s (2008)
  • Author(s): Mieko Yamada
  • Organizer: Combinatorics 2008
  • Place of Presentation: Costermano, Italy
  • Year and Date: 2008-06-24
• [Presentation] Difference sets over Galois rings GR(2^t, s)for evenn and evens (2008)
  • Author(s): Mieko Yamada(発表標題)
  • Organizer: Combinatorics
  • Place of Presentation: Costermano(Italy)
• [Presentation] Difference sets, divisible difference families and codes over Galois rings of characteristic 2^n
  • Author(s): Mieko Yamada
  • Organizer: Workshop on Combinatorial Designs
  • Place of Presentation: シンガポール（National University of Singapore）
  • Invited
• [Presentation] 標数2^nのガロア環の拡張BCH符号の部分符号について
  • Author(s): 大江純矢、山田美枝子
  • Organizer: 2011年応用数学合同研究集会
  • Place of Presentation: 滋賀（龍谷大学）
• [Presentation] The constructions of unreal Butson-type Hadamard matrices BH(n,6)'s
  • Author(s): Pritta Etriana Putri and Mieko Yamada
  • Organizer: 2011年応用数学合同研究集会
  • Place of Presentation: 滋賀（龍谷大学）
• [Book] International Symposium on Computational Science 2011
  • Author(s): Muhammad Ilyas and Mieko Yamada
  • Publisher: gakkotosho