Blocking sets of secant and tangent lines with respect to a quadric of $$\text{ PG }(n,q)$$

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2025-01-17 DOI:10.1007/s10623-024-01559-8

Bart De Bruyn, Puspendu Pradhan, Binod Kumar Sahoo

引用次数: 0

Abstract

For a set ${\mathcal {L}}$ of lines of $\text{ PG }(n,q)$, a set X of points of $\text{ PG }(n,q)$ is called an ${\mathcal {L}}$-blocking set if each line of ${\mathcal {L}}$ contains at least one point of X. Consider a possibly singular quadric Q of $\text{ PG }(n,q)$ and denote by ${\mathcal {S}}$ (respectively, ${\mathcal {T}}$) the set of all lines of $\text{ PG }(n,q)$ meeting Q in 2 (respectively, 1 or $q+1$) points. For ${\mathcal {L}}\in \{{\mathcal {S}},{\mathcal {T}}\cup {\mathcal {S}}\}$, we find the minimal cardinality of an ${\mathcal {L}}$-blocking set of $\text{ PG }(n,q)$ and determine all ${\mathcal {L}}$-blocking sets of that minimal cardinality.

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关于二次函数的正割线和切线的块集 $$\text{ PG }(n,q)$$

对于${\mathcal {L}}$的$\text{ PG }(n,q)$的线集合，如果${\mathcal {L}}$的每条线包含至少一个点X，则$\text{ PG }(n,q)$的点集合X称为${\mathcal {L}}$ -blocking set，考虑$\text{ PG }(n,q)$的一个可能的奇异二次型Q，用${\mathcal {S}}$（分别为${\mathcal {T}}$）表示$\text{ PG }(n,q)$的所有线的集合在2个（分别为1个或$q+1$）点中与Q相遇。对于${\mathcal {L}}\in \{{\mathcal {S}},{\mathcal {T}}\cup {\mathcal {S}}\}$，我们找到$\text{ PG }(n,q)$的${\mathcal {L}}$阻塞集的最小基数，并确定该最小基数的所有${\mathcal {L}}$阻塞集。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.