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A Geometric Study on Optimal Linear Codes Problem

Research Project

Project/Area Number 16K05256
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionOsaka Prefecture University

Principal Investigator

Maruta Tatsuya  大阪府立大学, 理学(系)研究科(研究院), 教授 (80239152)

Project Period (FY) 2016-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords線形符号 / 最適線形符号 / 符号の拡張可能性 / 有限射影幾何 / Griesmer 限界 / 最適符号 / blocking set / 有限幾何 / 符号の拡張
Outline of Final Research Achievements

A linear code with length n, dimension k and minimum weight d over the field of q elements is called an [n,k,d]q code. A classical and fundamental research problem in coding theory is to find n_q(k,d), the minimum length n for which an [n,k,d]q code exists, which is called "Optimal Linear Codes Problem".
In this work, considering the multiset consisting the columns of a generator matrix of a given code as a multiset in the projective space, we have obtained several new results on the problem by showing the nonexistence of some linear codes attaining the known bound using the geometric methods, constructing new codes and corresponding arcs and blocking sets in the projective spaces by the heuristic search with the aid of computers.

Academic Significance and Societal Importance of the Research Achievements

情報の電子化が進み、情報ネットワークが社会の基盤として不可欠となった現在、情報の伝送や記録の際に生じる誤りを正しく訂正できる確率が大きな誤り訂正符号を構成する必要がある。現在、実用化されている誤り訂正符号の多くは、Reed-Solomon 符号を代表とする有限体上の線形符号であるが、情報の伝達速度を左右する符号長や伝達できる情報量を示す次元を固定しても、誤り訂正能力の限界が確定できていない場合が多い。本研究は、線形符号を生成するために必要な射影空間上の集合を数学的に解析することにより、符号の誤り訂正能力の限界を解明しようとするものである。

Report

(6 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (48 results)

All 2021 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (4 results) Journal Article (15 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 13 results,  Open Access: 6 results,  Acknowledgement Compliant: 3 results) Presentation (26 results) (of which Int'l Joint Research: 13 results) Remarks (3 results)

  • [Int'l Joint Research] Institute of Mathematics and Informatics/Bulgarian Academy of Sciences(ブルガリア)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] Bulgarian Academy of Sciences(ブルガリア)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Gyeongsang National University(韓国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Gyeongsang National University(韓国)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Characteristic vector and weight distribution of a linear code2021

    • Author(s)
      I. Bouyukliev, S. Bouyuklieva, T. Maruta, P. Piperkov
    • Journal Title

      Cryptography and Communications

      Volume: 13 Issue: 2 Pages: 263-282

    • DOI

      10.1007/s12095-020-00458-8

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the (29,5)-Arcs in PG(2,7) and Some Generalized Arcs in PG(2,q)2020

    • Author(s)
      Iliya Bouyukliev, Eun Ju Cheon, Tatsuya Maruta, Tsukasa Okazaki
    • Journal Title

      Mathematics

      Volume: 8 Issue: 3 Pages: 320-320

    • DOI

      10.3390/math8030320

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Construction of new Griesmer codes of dimension 52019

    • Author(s)
      Y. Inoue, T. Maruta
    • Journal Title

      京都大学数理解析研究所講究録「代数系、論理、言語と計算機科学の周辺」

      Volume: 2130 Pages: 109-112

    • NAID

      120006724790

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] Nonexistence of some Griesmer codes of dimension 52019

    • Author(s)
      W. Kuranaka, T. Maruta
    • Journal Title

      京都大学数理解析研究所講究録「代数系、論理、言語と計算機科学の周辺」

      Volume: 2130 Pages: 113-118

    • NAID

      120006888051

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] Construction of new Griesmer codes of dimension 52019

    • Author(s)
      Y. Inoue, T. Maruta
    • Journal Title

      Finite Fields and their Applications

      Volume: 55 Pages: 231-237

    • DOI

      10.1016/j.ffa.2018.10.007

    • NAID

      120006724790

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Nonexistence of some 4-dimensional Griesmer codes over finite fields2018

    • Author(s)
      K. Kumegawa, T. Maruta
    • Journal Title

      Journal of Algebra Combinatorics Discrete Structures and Applications

      Volume: 5 Pages: 101-116

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] On the 3-extendability of quaternary linear codes2018

    • Author(s)
      H. Kanda, T. Maruta
    • Journal Title

      Finite Fields and their Applications

      Volume: 52 Pages: 126-136

    • DOI

      10.1016/j.ffa.2018.04.002

    • NAID

      120006493198

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On Critical Exponents of Dowling Matroids2018

    • Author(s)
      Y. Koga, T. Maruta, K. Shiromoto
    • Journal Title

      Designs, Codes and Cryptography

      Volume: 86 Issue: 9 Pages: 1947-1962

    • DOI

      10.1007/s10623-017-0431-8

    • NAID

      120006724788

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Nonexistence of some linear codes over the field of order four2018

    • Author(s)
      H. Kanda, T. Maruta
    • Journal Title

      Discrete Mathematics

      Volume: 341 Issue: 10 Pages: 2676-2685

    • DOI

      10.1016/j.disc.2018.06.024

    • NAID

      120006724789

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Construction of two-weight codes over finite fields and its applications2017

    • Author(s)
      E.J. Cheon, Y. Kageyama, S.J. Kim, N. Lee, T. Maruta
    • Journal Title

      Bulletin of the Korean Mathematical Society

      Volume: 54 Issue: 3 Pages: 731-736

    • DOI

      10.4134/bkms.b151011

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Some new 4-dimensional linear codes2017

    • Author(s)
      N. Bono, T. Maruta
    • Journal Title

      Proceedings of 8th International Workshop on Optimal Codes and Related Topics (OC 2017)

      Volume: 8

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Nonexistence of some Griesmer codes of dimension 4 over F_82017

    • Author(s)
      M. Fujii, T. Maruta
    • Journal Title

      Proceedings of 8th International Workshop on Optimal Codes and Related Topics (OC 2017)

      Volume: 8

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the minimum length of linear codes over the field of 9 elements2017

    • Author(s)
      K. Kumegawa, T. Okazaki, T. Maruta
    • Journal Title

      Electronic J. Combinatorics

      Volume: 24

    • NAID

      120006605036

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Construction of some triple blocking sets in PG(2,q)2016

    • Author(s)
      E.J. Cheon, T. Maruta, T. Okazaki
    • Journal Title

      Proceedings of 15th International Workshop on Algebraic and Combinatorial Coding Theory

      Volume: 15 Pages: 96-101

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] On the geometric constructions of optimal linear codes2016

    • Author(s)
      Y. Kageyama, T. Maruta
    • Journal Title

      Designs, Codes and Cryptography

      Volume: 81 Issue: 3 Pages: 469-480

    • DOI

      10.1007/s10623-015-0167-2

    • NAID

      120006312688

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] A Conjecture on Optimal Ternary Linear Codes2020

    • Author(s)
      D. Kawabata, T. Maruta
    • Organizer
      The 17th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT 2020)
    • Related Report
      2020 Annual Research Report
  • [Presentation] On the achievement of the Griesmer bound for ternary linear codes2020

    • Author(s)
      川端大輝, 丸田辰哉
    • Organizer
      離散数学とその応用研究集会2020 (JCCA2020)
    • Related Report
      2020 Annual Research Report
  • [Presentation] New linear codes over the field of order 72020

    • Author(s)
      野村啓太, 丸田辰哉
    • Organizer
      離散数学とその応用研究集会2020 (JCCA2020)
    • Related Report
      2020 Annual Research Report
  • [Presentation] Nonexistence of some ternary linear codes2019

    • Author(s)
      澤島利治, 丸田辰哉
    • Organizer
      離散数学とその応用研究集会2019
    • Related Report
      2019 Research-status Report
  • [Presentation] A new extension theorem for quaternary linear codes2019

    • Author(s)
      白水 誠, 丸田辰哉
    • Organizer
      離散数学とその応用研究集会2019
    • Related Report
      2019 Research-status Report
  • [Presentation] On the extendability of linear codes over the eld of order four2019

    • Author(s)
      T. Maruta, H. Kanda, M. Shirouzu
    • Organizer
      The 42nd Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (42ACCMCC)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Nonexistence of some ternary linear codes of dimension 62019

    • Author(s)
      T. Sawashima, T. Maruta
    • Organizer
      The 42nd Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (42ACCMCC)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the minimum length of linear codes of dimension five2019

    • Author(s)
      W. Kuranaka, T. Maruta
    • Organizer
      The 42nd Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (42ACCMCC)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the extendability of linear codes over the field of order four2019

    • Author(s)
      白水 誠, 苅田仁, 丸田辰哉
    • Organizer
      2019年度応用数学合同研究集会
    • Related Report
      2019 Research-status Report
  • [Presentation] On the construction of optimal linear codes2019

    • Author(s)
      丸田辰哉
    • Organizer
      組合せ論的符号理論
    • Related Report
      2018 Research-status Report
  • [Presentation] Nonexistence of some Griesmer codes of dimension 42019

    • Author(s)
      藤井摩耶, 丸田辰哉
    • Organizer
      組合せ論的符号理論
    • Related Report
      2018 Research-status Report
  • [Presentation] On the non-trivial blocking sets in binary projective spaces2019

    • Author(s)
      坊野奈波, 丸田辰哉
    • Organizer
      組合せ論的符号理論
    • Related Report
      2018 Research-status Report
  • [Presentation] Construction of new Griesmer codes of dimension 52019

    • Author(s)
      Y. Inoue, T. Maruta
    • Organizer
      代数系、論理、言語と計算機科学の周辺
    • Related Report
      2018 Research-status Report
  • [Presentation] Nonexistence of some Griesmer codes of dimension 52019

    • Author(s)
      W. Kuranaka, T. Maruta
    • Organizer
      代数系、論理、言語と計算機科学の周辺
    • Related Report
      2018 Research-status Report
  • [Presentation] Constructing new linear codes over the field of order five2018

    • Author(s)
      Y. Inoue, T. Maruta
    • Organizer
      Japanese Conference on Combinatorics and its Applications (JCCA 2018)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Nonexistence of some Griesmer codes over the field of order five2018

    • Author(s)
      W. Kuranaka, T. Maruta
    • Organizer
      Japanese Conference on Combinatorics and its Applications (JCCA 2018)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Extendability of quaternary linear codes of dimension three2018

    • Author(s)
      M. Shirouzu, T. Maruta
    • Organizer
      Japanese Conference on Combinatorics and its Applications (JCCA 2018)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Construction of new Griesmer codes of dimension 52018

    • Author(s)
      Y. Inoue, T. Maruta
    • Organizer
      The 41st Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (41ACCMCC)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] A new extension theorem for ternary linear codes2018

    • Author(s)
      H. Kanda, T. Maruta
    • Organizer
      The 41st Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (41ACCMCC)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the non-trivial blocking sets in binary projective spaces2018

    • Author(s)
      N. Bono, T. Maruta, K. Shiromoto, K. Yamada
    • Organizer
      The 41st Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (41ACCMCC)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Nonexistence of some Griesmer codes of dimension 4 over F_82017

    • Author(s)
      M. Fujii, T. Maruta
    • Organizer
      Proceedings of 8th International Workshop on Optimal Codes and Related Topics (OC 2017)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Some new 4-dimensional linear codes2017

    • Author(s)
      N. Bono, T. Maruta
    • Organizer
      Proceedings of 8th International Workshop on Optimal Codes and Related Topics (OC 2017)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Nonexistence of some linear codes over the field of order four2017

    • Author(s)
      T. Maruta
    • Organizer
      The 5th International Combinatorics Conference (5ICC)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Nonexistence of some linear codes over F_42017

    • Author(s)
      苅田仁, 丸田辰哉
    • Organizer
      2017年度応用数学合同研究集会
    • Related Report
      2017 Research-status Report
  • [Presentation] On the extendability of quaternary linear codes2016

    • Author(s)
      丸田辰哉, 苅田仁
    • Organizer
      2016年度応用数学合同研究集会
    • Place of Presentation
      龍谷大学(滋賀県大津市)
    • Year and Date
      2016-12-17
    • Related Report
      2016 Research-status Report
  • [Presentation] Construction of some triple blocking sets in PG(2,q)2016

    • Author(s)
      Eun Ju Cheon, Tatsuya Maruta, Tsukasa Okazaki
    • Organizer
      Fifteenth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT 2016)
    • Place of Presentation
      Malibu Hotel(Albena, ブルガリア)
    • Year and Date
      2016-06-19
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Remarks] Griesmer Bound for Linear Codes over Finite Fields

    • URL

      http://www.mi.s.osakafu-u.ac.jp/~maruta/

    • Related Report
      2020 Annual Research Report 2019 Research-status Report
  • [Remarks] Griesmer Bound for Linear Codes over Finite Fields

    • URL

      http://www.mi.s.osakafu-u.ac.jp/~maruta/griesmer/

    • Related Report
      2018 Research-status Report 2017 Research-status Report
  • [Remarks] Griesmer bound for linear codes over finite fi eld

    • URL

      http://www.mi.s.osakafu-u.ac.jp/~maruta/griesmer/

    • Related Report
      2016 Research-status Report

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Published: 2016-04-21   Modified: 2022-01-27  

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