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<p>In this paper, we characterize blocking sets of external lines, called external-blocking sets, of a subplane with respect to individual projective planes. We first give a general lower bound for the size of the external-blocking sets in symmetric designs. Next, we find out the existence of a perfect external-blocking set, that is, an external-blocking set which touches the given lower bound. Finally, we prove that for a Desarguesian plane <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(PG(2,q^4)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <msup> <mi>q</mi> <mn>4</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, a perfect external-blocking set of a subplane <i>PG</i>(2,&#xa0;<i>q</i>) exists, and for a Desarguesian plane <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(PG(2,p^3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <msup> <mi>p</mi> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p\ge 7\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>, we give a necessary and sufficient condition for the existence of a perfect external-blocking set of a subplane <i>PG</i>(2,&#xa0;<i>p</i>). For example, we give perfect external-blocking sets of the Fano subplane in a Hall plane, a non-Desarguesian projective plane of order 9.</p>

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Blocking sets of external lines to a subplane in projective planes

  • 期刊论文

  • 发表日期：

    2025年10月22日

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  • Wonhong Lee,
  • Sang-Mok Kim

摘要

In this paper, we characterize blocking sets of external lines, called external-blocking sets, of a subplane with respect to individual projective planes. We first give a general lower bound for the size of the external-blocking sets in symmetric designs. Next, we find out the existence of a perfect external-blocking set, that is, an external-blocking set which touches the given lower bound. Finally, we prove that for a Desarguesian plane \(PG(2,q^4)\) P G ( 2 , q 4 ) , a perfect external-blocking set of a subplane PG(2, q) exists, and for a Desarguesian plane \(PG(2,p^3)\) P G ( 2 , p 3 ) with \(p\ge 7\) p ≥ 7 , we give a necessary and sufficient condition for the existence of a perfect external-blocking set of a subplane PG(2, p). For example, we give perfect external-blocking sets of the Fano subplane in a Hall plane, a non-Desarguesian projective plane of order 9.