%0 Journal Article
%T Subplanes of <i>PG</i>(2,<i>q<sup>r</sup></i>), Ruled Varieties V<sup>2<i>r</i>-1</sup><sub style="margin-left:-38px;position: relative;top: 0.5em; ">2</sub>&nbsp;&nbsp;&nbsp;&nbsp;in&nbsp;<i>PG</i>( 2<i>r</i>,<i>q</i>), and Related Codes
%A Rita Vincenti
%J Open Journal of Discrete Mathematics
%P 54-71
%@ 2161-7643
%D 2024
%I Scientific Research Publishing
%R 10.4236/ojdm.2024.144006
%X In this note we consider ruled varieties <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>V</mi>
   <mn>2</mn>
   <mrow>
    <mn>2</mn><mi>r</mi><mo>&#x2212;</mo><mn>1</mn></mrow>
  </msubsup>
  </mrow>
</math>

 of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>P</mi><mi>G</mi><mrow><mo>(</mo>
   <mrow>
    <mn>2</mn><mi>r</mi><mo>,</mo><mi>q</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>

, generalizing some results shown for <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>r</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow>
</math>

 in previous papers. By choosing appropriately two directrix curves, a <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>V</mi>
   <mn>2</mn>
   <mrow>
    <mn>2</mn><mi>r</mi><mo>&#x2212;</mo><mn>1</mn></mrow>
  </msubsup>
  </mrow>
</math>

 represents a non-affine subplane of order <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>q</mi>
</math>

 of the projective plane <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>P</mi><mi>G</mi><mrow><mo>(</mo>
   <mrow>
    <mn>2</mn><mo>,</mo><msup>
     <mi>q</mi>
     <mi>r</mi>
    </msup>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>

 represented in <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>P</mi><mi>G</mi><mrow><mo>(</mo>
   <mrow>
    <mn>2</mn><mi>r</mi><mo>,</mo><mi>q</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>

 by a spread of a hyperplane. That proves the conjecture assumed in [1]. Finally, a large family of linear codes dependent on <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>r</mi><mo>&#x2265;</mo><mn>2</mn></mrow>
</math>

 is associated with projective systems defined both by <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>V</mi>
   <mn>2</mn>
   <mrow>
    <mn>2</mn><mi>r</mi><mo>&#x2212;</mo><mn>1</mn></mrow>
  </msubsup>
  </mrow>
</math>

 and by a maximal bundle of such varieties with only an <i>r</i>-directrix in common, then are shown their basic parameters.
%K Finite Geometry
%K Translation Planes
%K Spreads
%K Varieties
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=136818