(2n, 2n, 2n, 1)‐Relative Difference Sets and Their Representations. Issue 12 (16th April 2013)

• Record Type: Journal Article
• Authors: Zhou, Yue
• Is Part Of: Journal of combinatorial designs. Volume 21:Issue 12(2013:Dec.)
• Page Start: 563
• Page End: 584
• Publication Date: 2013-04-16
• Subjects: Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21349
• Languages: English
• ISSNs: 1063-8539

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We show that every <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25szw3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21349:jcd21349-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo>, </mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo>, </mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo>, </mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives>‐relative difference set <italic>D</italic> in <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25szxn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21349:jcd21349-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>4</mml:mn><mml:mi>n</mml:mi></mml:msubsup></mml:math></alternatives> relative to <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25szsf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21349:jcd21349-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msubsup></mml:math></alternatives> can be represented by a polynomial <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25szt0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21349:jcd21349-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant="double-struck">F</mml:mi><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mi>x</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives>, where <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t03d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21349:jcd21349-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>a</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:math></alternatives> is a permutation for each nonzero <italic>a</italic>. We call such an <italic>f</italic> a planar function on <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t062" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21349:jcd21349-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="double-struck">F</mml:mi><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:msub></mml:math></alternatives>. The projective plane Π obtained from <italic>D</italic> in the way of M. J. Ganley and E. Spence (J Combin Theory Ser A, 19(2) (1975), 134–153) is coordinatized, and we obtain necessary and sufficient conditions of Π to be a presemifield plane. We also prove that a function <italic>f</italic> on <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t00r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21349:jcd21349-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="double-struck">F</mml:mi><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:msub></mml:math></alternatives> with exactly two elements in its image set and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t3fg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21349:jcd21349-math-0010" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives> is planar, if and only if, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t3dx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21349:jcd21349-math-0011" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives> for any <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t3hk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21349:jcd21349-math-0012" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mo>, </mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant="double-struck">F</mml:mi><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:msub></mml:mrow></mml:math></alternatives>.</p> </abstract>