Conics arising from external points and their binary codes

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• 作者：Adonus L. Madison ; Junhua Wu
• 关键词：Conic ; External points ; Incidence matrix ; Orbit ; Dimension
• 刊名：Designs, Codes and Cryptography
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：78
• 期：2
• 页码：473-491
• 全文大小：489 KB
• 参考文献：1.Abatangelo V., Fisher J.C., Korchm\(\acute{a}\) ros G., Larato B.: On the mutual position of two irreducible conics in \(\text{ PG }(2, q)\) , \(q\) odd. Adv. Geom. 11(4), 604–613 (2011).
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  9.Madison A.L., Wu J.: On binary codes from conics in \(\text{ PG }(2, q)\) . Eur. J. Combin. 33, 33–48 (2012).
  10.Wu J.: Conics arising from internal points and their binary codes. Linear Algebra Appl. 439, 422–434 (2013).
  
• 作者单位：Adonus L. Madison (1)
  Junhua Wu (2)
  
  1. Lane College, Jackson, TN, USA
  2. Department of Mathematics, Lane College, Jackson, TN, USA
  
• 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

In Wu (Linear Algebra Appl 439:422–434 2013), the author constructed a binary code using the incidence matrix of conics consisting only of internal points with respect to a fixed conic versus internal points and studied geometric problems associated with this code. Inspired by that work, in this article, we construct conics consisting only of external points with respect to a conic for \(q\) odd. We study the intersection pattern of each of these conics with secant lines of the fixed conic, compute the dimension of the \({\mathbb F}_2\)-row space of the incidence matrix of the aforementioned conics and external points which provides us with the dimension of the associated binary code, and find the automorphism group of the binary code.