Algebraic constructions for combinatorial designs and their applications to combinatorial testing

2021 Fiscal Year Final Research Report

• Project/Area Number: 19K14585
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12030:Basic mathematics-related
• Research Institution: University of Yamanashi
• Principal Investigator: Lu Xiao-Nan 山梨大学, 大学院総合研究部, 特任助教 (10805683)
• Project Period (FY): 2019-04-01 – 2022-03-31
• Keywords: Combinatorial design / Combinatorial testing / Locating array / Orthogonal array / Steiner quadruple system / Fault location / Adaptive algorithm / Error-correcting code

Outline of Final Research Achievements

Focusing on algebraic constructions of combinatorial designs, this research is devoted for clarifying the relationship among different algebraic constructions, and exploring the applications of those combinatorial designs. In particular, for combinatorial interaction testing, both new constructions of combinatorial arrays and improved faulty location algorithms are developed. Main contributions include: (1) Generalization of mutually orthogonal Latin squares to higher dimensions; (2) Proof of existence of abelian-group invariant Steiner quadruple systems; (3) Algebraic characterization, statical optimality, new results and classifications by computer search for circulant almost orthogonal arrays; (4) Clarifying the linear dimensions of a class of BCH codes with large distance. (5) Proposing a new bound for locating arrays. (6) Improving adaptive algorithms for fault location in combinatorial testing; (7) Introducing the notion and constructions of error-correcting locating arrays.

Free Research Field

離散数学，数理情報学

Academic Significance and Societal Importance of the Research Achievements

本研究では，組合せデザインと呼ばれる離散構造の代数的側面に焦点を当てて，それらの内在する関連性の解明およびそれらの符号および統計的実験計画への応用について研究を行って，各問題に新たな研究結果を得た．また，複数のコンポーネントが組み込まれる複雑システムにおける故障を検出するための数理モデルとして，組合せデザイン・代数学・符号理論等多様な手法を用いて，検査計画問題の理論的限界のより精確な評価を与え，検出アルゴリズムの効率化に成功した．本研究で得られた成果は，情報通信・実験計画・ソフトウェア工学等の領域において，基礎数学理論・数理モデルおよび関連するアルゴリズムを貢献することになる．