Construction of high-performance error-correcting codes using Grobner bases

• Project/Area Number: 15K13994
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Communication/Network engineering
• Research Institution: Toyota Technological Institute
• Principal Investigator: Matsui Hajime 豊田工業大学, 工学(系)研究科(研究院), 准教授 (80329854)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2015: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
• Keywords: 離散フーリエ変換 / 多値論理関数 / 準巡回符号 / 離散Fourier変換 / ユークリッド整域 / 畳み込み定理 / 整数符号 / 代数的符号 / 符号化 / 復号化 / リード・マラー符号 / LFSR

Outline of Final Research Achievements

1. A fast decoding method of projective Reed-Muller codes has been established (collaboration with Dr. Norihiro Nakashima). Compared to decoding using the Gaussian elimination method, the order of its computational complexity could be reduced.
2. The error-correcting codes over the residue rings of the Euclidean domains have been investigated. They include quasi-cyclic codes and integer codes as special cases. We showed that we can uniquely determine the generator matrix for each code. This fact is useful for code construction and search.
3. The convolution theorem for a class of multiple-valued logic polynomials has been established and a fast calculation method for the multiplication of them has been derived. We showed that the discrete Fourier transform used in the convolution theorem has a transposition relation to that used for decoding affine variaty codes.

Research Products

• [Int'l Joint Research] アイン・シャムス大学(エジプト)
• [Journal Article] Multiplicative structure and Hecke rings of generator matrices for codes over quotient rings of Euclidean domains (2017)
  • Author(s): Hajime Matsui
  • Journal Title: MDPI Mathematics Volume: vol.5, no.4, 82 Issue: 4 Pages: 82-82
  • DOI: 10.3390/math5040082
  • Peer Reviewed / Open Access
• [Journal Article] A Convolution Theorem for Multiple-Valued Logic Polynomials of a Semigroup Type and Their Fast Multiplication (2016)
  • Author(s): Hajime Matsui
  • Journal Title: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences Volume: E99.A Issue: 6 Pages: 1025-1033
  • DOI: 10.1587/transfun.E99.A.1025
  • NAID: 130005154211
  • ISSN: 0916-8508, 1745-1337
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces (2016)
  • Author(s): Norihiro Nakashima, Hajime Matsui
  • Journal Title: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences Volume: E99.A Issue: 3 Pages: 733-741
  • DOI: 10.1587/transfun.E99.A.733
  • NAID: 130005131283
  • ISSN: 0916-8508, 1745-1337
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] 多値論理多項式に対する離散フーリエ変換の応用と積の高速化 (2018)
  • Author(s): 松井一
  • Organizer: 電子情報通信学会 情報理論研究会
  • Invited
• [Presentation] 論理多項式を用いたベイズ学習における漸近評価の分類 (2017)
  • Author(s): 市川翔太，井上直樹，椋野純一，松井一
  • Organizer: 第40回情報理論とその応用シンポジウム
• [Presentation] On multiple-valued logic polynomials of a subset type (2016)
  • Author(s): Hajime Matsui
  • Organizer: Recent Results Posters at The International Symposium on Information Theory and Its Applications (ISITA2016)
  • Place of Presentation: Monterey, California, USA
  • Year and Date: 2016-10-30
  • Int'l Joint Research
• [Presentation] Garcia-Stichtenothによる代数曲線符号の誤り訂正計算量の削減 (2016)
  • Author(s): 中島規博，松井一
  • Organizer: 日本数学会秋季総合分科会 応用数学分科会
  • Place of Presentation: 関西大学千里山キャンパス（大阪府吹田市）
  • Year and Date: 2016-09-15
• [Presentation] Modified DFTs for affine variety codes (2015)
  • Author(s): Norihiro Nakashima, Hajime Matsui
  • Organizer: 第38回情報理論とその応用シンポジウム
  • Place of Presentation: 下電ホテル（岡山県倉敷市）
  • Year and Date: 2015-11-24
• [Presentation] On multiple-valued logic polynomials of a product type (2015)
  • Author(s): Hajime Matsui
  • Organizer: 第38回情報理論とその応用シンポジウム
  • Place of Presentation: 下電ホテル（岡山県倉敷市）
  • Year and Date: 2015-11-24
• [Presentation] グレブナー基底とDFTを用いたエルミート曲線符号の符号化・復号化 (2015)
  • Author(s): 中島規博，松井一
  • Organizer: 電気・電子・情報関係学会東海支部連合大会
  • Place of Presentation: 名古屋工業大学（愛知県名古屋市）
  • Year and Date: 2015-09-28
• [Presentation] ガウス素数の掘割問題についての虚二次数体への一般化と右手法 (2015)
  • Author(s): 木下真志，松井一
  • Organizer: 電気・電子・情報関係学会東海支部連合大会
  • Place of Presentation: 名古屋工業大学（愛知県名古屋市）
  • Year and Date: 2015-09-28
• [Presentation] 右手法を用いた虚二次数体の素元における掘割探索法の並列化 (2015)
  • Author(s): 木下真志，松井一
  • Organizer: 電子情報通信学会ソサイエティ大会
  • Place of Presentation: 東北大学（宮城県仙台市）
  • Year and Date: 2015-09-08
• [Presentation] 有限体の部分半群におけるDFTのアフィン多様体符号への応用 (2015)
  • Author(s): 中島規博，松井一
  • Organizer: 電子情報通信学会ソサイエティ大会
  • Place of Presentation: 東北大学（宮城県仙台市）
  • Year and Date: 2015-09-08
• [Presentation] 符号理論と離散フーリエ変換 (2015)
  • Author(s): 松井一
  • Organizer: 第4回誤り訂正符号のワークショップ
  • Place of Presentation: 白山菖蒲亭（石川県加賀市）
  • Year and Date: 2015-09-02
• [Presentation] A semigroup DFT over finite fields applied to affine variety codes (2015)
  • Author(s): Norihiro Nakashima, Hajime Matsui
  • Organizer: Recent Results Session at the 2015 IEEE International Symposium on Information Theory (ISIT 2015)
  • Place of Presentation: Hong Kong (China)
  • Year and Date: 2015-06-14
  • Int'l Joint Research
• [Presentation] 虚二次数体の素元に対する掘割問題についての高速探索法 (2015)
  • Author(s): 木下真志，松井一
  • Organizer: 電子情報通信学会 技術研究報告（コンピュテーション研究会）
  • Place of Presentation: 定山渓ビューホテル（北海道札幌市）
  • Year and Date: 2015-06-12
• [Remarks] 豊田工業大学情報通信研究室
  • URL: https://www.toyota-ti.ac.jp/Lab/Denshi/InfoComm/index_ja.html
• [Remarks] 豊田工業大学研究者情報システム
  • URL: http://ttiweb.toyota-ti.ac.jp/public/user.php
• [Remarks] 豊田工業大学情報通信研究室
  • URL: http://www.toyota-ti.ac.jp/Lab/Denshi/InfoComm/index_ja.html