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• 作者：Michael Braun
• 关键词：\(q\) ; Analog ; \(t\) ; Wise balanced design ; Kramer–Mesner ; Borel subgroup
• 刊名：Designs, Codes and Cryptography
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：78
• 期：2
• 页码：383-390
• 全文大小：391 KB
• 参考文献：1.Borel A.: Groupes linéaires algébriques. Ann. Math. 64(1), 20–82 (1956).
  2.Braun M.: An algebraic interpretation of the \(q\) -binomial coefficients. Int. Electron. J. Algebr. 6, 23–30 (2009).
  3.Braun M., Kerber A., Laue R.: Systematic construction of \(q\) -analogs of designs. Des. Codes Cryptogr. 34(1), 55–70 (2005).
  4.Braun M., Etzion T., Östergård P. R. J., Vardy A., Wassermann A.: On the existence of \(q\) -analogs of steiner systems (submitted).
  5.Fazeli A., Lovett S., Vardy A.: Nontrivial \(t\) -designs over finite fields exist for all \(t\) . arXiv:​1306.​2088 , 2013.
  6.Itoh T.: A new family of \(2\) -designs over \(GF(q)\) admitting \(SL_m(q^l)\) . Geom. Dedicata 69, 261–286 (1998).
  7.Kramer E., Mesner D.: \(t\) -Designs on hypergraphs. Discret. Math. 15(3), 263–296 (1976).
  8.Miyakawa M., Munemasa A., Yoshiara S.: On a class of small \(2\) -designs over \(GF(q)\) . J. Comb. Des. 3, 61–77 (1995).
  9.Suzuki H.: \(2\) -designs over \(GF(2^m)\) . Graphs Comb. 6, 293–296 (1990).
  10.Suzuki H.: \(2\) -designs over \(GF(q)\) . Graphs Comb. 8, 381–389 (1992).
  11.Thomas S.: Designs over finite fields. Geom. Dedicata 24, 237–242 (1987).
  12.Thomas S.: Designs and partial geometries over finite fields. Geom. Dedicata 63, 247–253 (1996).
  
• 作者单位：Michael Braun (1)
  
  1. Faculty of Computer Science, University of Applied Sciences Darmstadt, Darmstadt, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Coding and Information Theory
  Data Structures, Cryptology and Information Theory
  Data Encryption
  Discrete Mathematics in Computer Science
  Information, Communication and Circuits
  
• 出版者：Springer Netherlands
• ISSN：1573-7586

文摘

A \(t\text {-}(n, K,\lambda ; q)\) design, also called the \(q\)-analog of a \(t\)-wise balanced design, is a set \({\mathcal {B}}\) of subspaces with dimensions contained in \(K\) of the \(n\)-dimensional vector space \({\mathbb {F}}_q^n\) over the finite field with \(q\) elements such that each \(t\)-subspace of \({\mathbb {F}}_q^n\) is contained in exactly \(\lambda \) elements of \({\mathcal {B}}\). In this paper we give a construction of an infinite series of nontrivial \(t\text {-}(n, K,\lambda ; q)\) designs for all dimensions \(t\ge 1\) and all prime powers \(q\) admitting the standard Borel subgroup as group of automorphisms. Furthermore, replacing \(q=1\) gives an ordinary \(t\)-wise balanced design defined on sets.