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A study of combinatorics over Galois rings

Research Project

Project/Area Number 24540013
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionTokyo Woman's Christian University (2014)
Kanazawa University (2012-2013)

Principal Investigator

YAMADA Mieko  東京女子大学, 現代教養学部, 研究員 (70130226)

Co-Investigator(Kenkyū-buntansha) MOMIHARA Koji  熊本大学, 教育学部, 講師 (70613305)
Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
Fiscal Year 2014: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywordsガロア環 / 差集合 / ガウス和 / Hadamard行列 / difference family / 符号 / 符号理論 / アダマール行列
Outline of Final Research Achievements

We gave a new approach and a new perspective on the study of combinatorics, that is,recognizing a combinatorial structure over finite fields and Galois rings as an image of a subset of a local field by natural projections. We obtained an infinite family of Menon-Hadamard difference sets over Galois rings of characteristic a power of 2 from a subset of 2-adic field by natural projections, which has a nested structure. Furthermore we showed that there exist difference families and Hadamard matrices over Galois rings of characteristic 4. For odd characteristics, we constructed a decoding algorithm of BCH codes with at most 2 errors. For a characteristic a square of a prime and an extension degree 2, we constructed difference families and Hadamard matrices.
In addition to the above results, we constructed Cayley graphs and their lifting and provided a new insight of the classification of skew Hadamard matrices over finite fields.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (40 results)

All 2015 2014 2013 2012 Other

All Journal Article (16 results) (of which Peer Reviewed: 16 results,  Acknowledgement Compliant: 4 results,  Open Access: 3 results) Presentation (21 results) (of which Invited: 7 results) Remarks (3 results)

  • [Journal Article] Menon-Hadamard difference sets obtained from a local field by natural projections2015

    • Author(s)
      M. Yamada
    • Journal Title

      Springer Proceedings in Mathematics and Statistics(Algebraic Design Theory and Hadamard Matrices)

      Volume: 印刷中

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Nonsymmetric primitive translation schemes of prime power order2015

    • Author(s)
      T. Feng, K. Momihara
    • Journal Title

      Journal of Algebraic Combinatorics

      Volume: 41 Issue: 1 Pages: 1-20

    • DOI

      10.1007/s10801-014-0523-8

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes2015

    • Author(s)
      T. Feng, K. Momihara, Q. Xiang
    • Journal Title

      Combinatorica

      Volume: 印刷中 Issue: 4 Pages: 413-434

    • DOI

      10.1007/s00493-014-2895-8

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Cameron-Liebler line classes with parameter $x=\frac{q^2-1}{2}$2015

    • Author(s)
      T. Feng, K. Momihara, Q. Xiang
    • Journal Title

      Journal of Combinatorial Theory, Series A

      Volume: 133 Pages: 307-338

    • DOI

      10.1016/j.jcta.2015.02.004

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Divisible difference family from Galois rings GR(4,n) and Hadamard matrices2014

    • Author(s)
      K. Momihara, M. Yamada
    • Journal Title

      Designs, Codes and Cryptography

      Volume: 73 Issue: 3 Pages: 897-909

    • DOI

      10.1007/s10623-013-9833-4

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Certain strongly regular Cayley graphs on F_{2^{2(2s+1)}} from cyclotomy2014

    • Author(s)
      K. Momihara
    • Journal Title

      Finite Fields and Their Applications

      Volume: 25 Pages: 280-292

    • DOI

      10.1016/j.ffa.2013.10.006

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Lifting constructions of strongly regular Cayley graphs2014

    • Author(s)
      K. Momihara, Q. Xiang
    • Journal Title

      Finite Fields and Their Applications

      Volume: 26 Pages: 86-99

    • DOI

      10.1016/j.ffa.2013.11.003

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Unified combinatorial constructions of optimal optical orthogonal codes2014

    • Author(s)
      C. Fan, K. Momihara
    • Journal Title

      Advances in Mathematics of Communications

      Volume: 8 Issue: 1 Pages: 53-66

    • DOI

      10.3934/amc.2014.8.53

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Difference sets overGalois rings with odd extension degree and characteristic an even power of 22013

    • Author(s)
      Mieko Yamada
    • Journal Title

      Designs, Codes and Cryptography

      Volume: 67 Issue: 1 Pages: 37-57

    • DOI

      10.1007/s10623-011-9584-z

    • Related Report
      2013 Research-status Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Three-class association schemes from cyclotomy2013

    • Author(s)
      K. Momihara
    • Journal Title

      Journal of Combinatorial Theory, Series A

      Volume: 120 Issue: 6 Pages: 1202-1215

    • DOI

      10.1016/j.jcta.2013.03.002

    • Related Report
      2013 Research-status Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Skew Hadamard difference sets from cyclotomic strongly regular graphs2013

    • Author(s)
      K. Momihara
    • Journal Title

      SIAM Journal on Discrete Mathematics

      Volume: 27 Issue: 2 Pages: 1112-1122

    • DOI

      10.1137/120888788

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Evaluation of the weight distribution of a class of cyclic codes based on index 2 Gauss sums2013

    • Author(s)
      T. Feng, K. Momihara
    • Journal Title

      IEEE Transactions on Information Theory

      Volume: 59 Issue: 9 Pages: 5980-5984

    • DOI

      10.1109/tit.2013.2259538

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Inequivalence of skew Hadamard difference sets and triple intersection numbers modulo a prime2013

    • Author(s)
      K. Momihara
    • Journal Title

      Electronic Journal of Combinatorics

      Volume: 20-P35 Pages: 1-19

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Strongly regular Cayley graphs, skew Hadamard difference sets, and rationality of relative Gauss sums2013

    • Author(s)
      K. Momihara
    • Journal Title

      European Journal of Combinatorics

      Volume: 34 Issue: 4 Pages: 706-723

    • DOI

      10.1016/j.ejc.2012.10.006

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Reed-Muller codes over Galois rings of cgaracteristic 2^n2012

    • Author(s)
      Soryo Kawasaki, Mieko Yamada
    • Journal Title

      Science Reports of Kanazawa University

      Volume: 56 Pages: 1-13

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] A decomposition of the 2-design formed by the planes in AG(2n,3)2012

    • Author(s)
      Koji Momihara, Miwako Mishima, Masakazu Jimbo
    • Journal Title

      Finite Fields and Their Applications

      Volume: Vol.18 Issue: 5 Pages: 956-970

    • DOI

      10.1016/j.ffa.2012.04.001

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] 古典的な指標和とそれに関連する組合せ論2015

    • Author(s)
      籾原幸二
    • Organizer
      第11回組合せ論若手研究集会
    • Place of Presentation
      慶應義塾大学(神奈川県・横浜市)
    • Year and Date
      2015-03-04 – 2015-03-05
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Three-valued Gauss periods and related designs and association schemes2014

    • Author(s)
      籾原幸二
    • Organizer
      RIMS研究集会「有限群とその表現, 頂点作用素代数, 代数的組合せ論の研究」
    • Place of Presentation
      京都大学(京都府・京都市)
    • Year and Date
      2014-12-16 – 2014-12-19
    • Related Report
      2014 Annual Research Report
  • [Presentation] Menon-Hadamard difference sets obtained from a local field by natural projections2014

    • Author(s)
      山田美枝子
    • Organizer
      研究集会「実験計画法およびその周辺の組合せ構造2014」
    • Place of Presentation
      城崎国際アートセンター(兵庫県・豊岡市)
    • Year and Date
      2014-12-13 – 2014-12-15
    • Related Report
      2014 Annual Research Report
  • [Presentation] New projective two-intersection sets and related Hadamard difference sets2014

    • Author(s)
      籾原幸二
    • Organizer
      研究集会「実験計画法およびその周辺の組合せ構造2014」
    • Place of Presentation
      城崎国際アートセンター(兵庫県・豊岡市)
    • Year and Date
      2014-12-13 – 2014-12-15
    • Related Report
      2014 Annual Research Report
  • [Presentation] Menon-Hadamard difference sets obtained from a local field by natural projections2014

    • Author(s)
      山田美枝子
    • Organizer
      Workshop on Hadamard Matrices and Combinatorial Designs
    • Place of Presentation
      東北大学(宮城県・仙台市)
    • Year and Date
      2014-10-31 – 2014-11-01
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Menon-Hadamard difference sets obtained from a local field by natural projections2014

    • Author(s)
      Mieko Yamada
    • Organizer
      Workshop on Algebraic Desighn Theory and Hadamard Matrices 2014
    • Place of Presentation
      Lethbridge大学(Canada)
    • Year and Date
      2014-07-08 – 2014-07-11
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 局所体の部分集合から得られるガロア環の差集合2014

    • Author(s)
      山田美枝子
    • Organizer
      第31回代数的組合せ論シンポジウム
    • Place of Presentation
      東北大学(宮城県・仙台市)
    • Year and Date
      2014-06-19 – 2014-06-20
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Cyclotomic schemes and related problems2014

    • Author(s)
      籾原幸二
    • Organizer
      代数的組合せ論「夏の学校2014」
    • Place of Presentation
      ホテルクレセント(宮城県・仙台市)
    • Year and Date
      2014-06-15 – 2014-06-18
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] skew Hadamard difference set とその非同値性について

    • Author(s)
      籾原幸二
    • Organizer
      離散数学とその応用研究集会
    • Place of Presentation
      山形(山形市保健センター)
    • Related Report
      2013 Research-status Report
  • [Presentation] Skew Hadamard difference setの非同値性の問題について

    • Author(s)
      籾原幸二
    • Organizer
      第10回「代数学と計算」研究集会
    • Place of Presentation
      東京(首都大学東京)
    • Related Report
      2013 Research-status Report
  • [Presentation] ガロア環上のBCH codeの復号アルゴリズム

    • Author(s)
      皆川泰蔵, 山田美枝子
    • Organizer
      応用数学合同研究集会
    • Place of Presentation
      滋賀(龍谷大学)
    • Related Report
      2013 Research-status Report
  • [Presentation] 局所体の部分集合から得られるガロア環の差集合

    • Author(s)
      山田美枝子
    • Organizer
      応用数学合同研究集会
    • Place of Presentation
      滋賀(龍谷大学)
    • Related Report
      2013 Research-status Report
  • [Presentation] Inequivalence of skew Hadamard difference sets

    • Author(s)
      K. Momihara
    • Organizer
      The 3rd Taiwan-Japan Conference on Combinatorics and its Applications
    • Place of Presentation
      Taiwan(National Chiayi University)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] A decoding algorithm of BCH codes over Galois rings

    • Author(s)
      Taizo Minagawa, Mieko Yamada
    • Organizer
      The 3rd Taiwan-Japan Conference on Combinatorics and its Applications
    • Place of Presentation
      Taiwan(National Chiayi University)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 強正則グラフCay(F_q,D)と相対ガウス和の有理性について

    • Author(s)
      籾原幸二
    • Organizer
      RIMS 研究集会「デザイン、符号、グラフおよびその周辺」
    • Place of Presentation
      京都(京都大学)
    • Related Report
      2012 Research-status Report
  • [Presentation] Strongly regular Cayley graphs and rationality of relative Gauss sums

    • Author(s)
      K. Momihara
    • Organizer
      Combinatorics 2012
    • Place of Presentation
      イタリア(ペルージャ)
    • Related Report
      2012 Research-status Report
  • [Presentation] Lifting construction of strongly regular Cayley graphs in F_q

    • Author(s)
      K. Momihara
    • Organizer
      The 2nd Japan-Taiwan Conference on Combinatorics and its Applications
    • Place of Presentation
      愛知(名古屋大学)
    • Related Report
      2012 Research-status Report
  • [Presentation] Z_p^2上のPreparata xodeの復号アルゴリズム

    • Author(s)
      皆川泰蔵、山田美枝子
    • Organizer
      2012年 応用数学合同研究集会
    • Place of Presentation
      滋賀(龍谷大学)
    • Related Report
      2012 Research-status Report
  • [Presentation] 標数4のガロア環上のkerdock code から得られるassociation scheme

    • Author(s)
      河﨑宗亮、山田美枝子
    • Organizer
      2012年 応用数学合同研究集会
    • Place of Presentation
      滋賀(龍谷大学)
    • Related Report
      2012 Research-status Report
  • [Presentation] Lifting constructions of strongly regular Cayley graphs

    • Author(s)
      K. Momihara
    • Organizer
      研究集会「代数的グラフ理論、スペクトラルグラフ理論および周辺領域」
    • Place of Presentation
      愛知(名古屋大学)
    • Related Report
      2012 Research-status Report
  • [Presentation] Lifting constructions of strongly regular graphs and association schemes in F_q

    • Author(s)
      K. Momihara
    • Organizer
      RIMS 研究集会「有限群とその表現,頂点作用素代数,代数的組合せ論の研究」
    • Place of Presentation
      京都(京都大学)
    • Related Report
      2012 Research-status Report
  • [Remarks] 熊本大学ホームページ

    • URL

      http://www.educ.kumamoto-u.ac.jp/~momihara/

    • Related Report
      2014 Annual Research Report
  • [Remarks] 熊本大学応用数学研究室

    • URL

      http://combin-math-kumamoto.jp/

    • Related Report
      2013 Research-status Report
  • [Remarks] Web page of Koji Momihara

    • URL

      http://www.educ.kumamoto-u.ac.jp/~momihara/

    • Related Report
      2013 Research-status Report

URL: 

Published: 2013-05-31   Modified: 2019-07-29  

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