Group Divisible Designs with Block Size Four and Group Type gum1 for g≡0mod6. Issue 1 (10th December 2012)

• Record Type: Journal Article
• Authors: Wei, Hengjia; Ge, Gennian
• Is Part Of: Journal of combinatorial designs. Volume 22:Issue 1(2014:Jan.)
• Page Start: 26
• Page End: 52
• Publication Date: 2012-12-10
• Subjects: Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21336
• Languages: English
• ISSNs: 1063-8539

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Nonuniform group divisible designs (GDDs) have been studied by numerous researchers for the past two decades due to their essential role in the constructions for other types of designs. In this paper, we investigate the existence problem of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbwxb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:mn>4</mml:mn><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives>‐GDDs of type <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbwtp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mi>u</mml:mi></mml:msup><mml:msup><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math></alternatives> for <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbx22" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>g</mml:mi><mml:mo>≡</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.28em" /><mml:mo form="prefix">mod</mml:mo><mml:mspace width="0.28em" /><mml:mn>6</mml:mn></mml:mrow></mml:math></alternatives>. First, we determine completely the spectrum of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbwzw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:mn>4</mml:mn><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives>‐GDDs of types <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbws4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mn>18</mml:mn><mml:mi>u</mml:mi></mml:msup><mml:msup><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math></alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbwmc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0010" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mn>36</mml:mn><mml:mi>u</mml:mi></mml:msup><mml:msup><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math></alternatives>. Furthermore, for general cases, we show that for each <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbwpg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0011" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>g</mml:mi><mml:mo>≡</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.28em" /><mml:mo form="prefix">mod</mml:mo><mml:mspace width="0.28em" /><mml:mn>6</mml:mn></mml:mrow></mml:math></alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbwj8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0012" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>g</mml:mi><mml:mo>≥</mml:mo><mml:mn>12</mml:mn></mml:mrow></mml:math></alternatives>, a <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbwfm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0013" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:mn>4</mml:mn><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives>‐GDD of type <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbwd2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0014" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mi>u</mml:mi></mml:msup><mml:msup><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math></alternatives> exists if and only if <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbw9d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0015" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math></alternatives>, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbw8v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0016" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo>≡</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.28em" /><mml:mo form="prefix">mod</mml:mo><mml:mspace width="0.28em" /><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbw56" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0017" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>m</mml:mi><mml:mo>≤</mml:mo><mml:mi>g</mml:mi><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives>, except possibly for <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbw4n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0018" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>g</mml:mi><mml:mo>≡</mml:mo><mml:mn>6</mml:mn><mml:mo>, </mml:mo><mml:mn>42</mml:mn><mml:mo>, </mml:mo><mml:mn>66</mml:mn><mml:mo>, </mml:mo><mml:mn>78</mml:mn><mml:mo>, </mml:mo><mml:mn>102</mml:mn><mml:mo>, </mml:mo><mml:mn>114</mml:mn><mml:mo>, </mml:mo><mml:mn>138</mml:mn><mml:mo>, </mml:mo><mml:mn>174</mml:mn><mml:mspace width="0.28em" /><mml:mo form="prefix">mod</mml:mo><mml:mspace width="0.28em" /><mml:mn>180</mml:mn></mml:mrow></mml:math></alternatives>, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbw10" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0019" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>g</mml:mi><mml:mo>≥</mml:mo><mml:mn>42</mml:mn></mml:mrow></mml:math></alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3t1kbtj7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:jcd21336:equation:jcd21336-math-0020" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>, </mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo><mml:mo>∈</mml:mo><mml:mo>{</mml:mo><mml:mo>(</mml:mo><mml:mn>7</mml:mn><mml:mo>, </mml:mo><mml:mn>3</mml:mn><mml:mi>g</mml:mi><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo><mml:mo>, </mml:mo><mml:mo>(</mml:mo><mml:mn>11</mml:mn><mml:mo>, </mml:mo><mml:mn>5</mml:mn><mml:mi>g</mml:mi><mml:mo>−</mml:mo><mml:mn>9</mml:mn><mml:mo>)</mml:mo><mml:mo>, </mml:mo><mml:mo>(</mml:mo><mml:mn>11</mml:mn><mml:mo>, </mml:mo><mml:mn>5</mml:mn><mml:mi>g</mml:mi><mml:mo>−</mml:mo><mml:mn>6</mml:mn><mml:mo>)</mml:mo><mml:mo>, </mml:mo><mml:mo>(</mml:mo><mml:mn>11</mml:mn><mml:mo>, </mml:mo><mml:mn>5</mml:mn><mml:mi>g</mml:mi><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo><mml:mo>, </mml:mo><mml:mo>(</mml:mo><mml:mn>13</mml:mn><mml:mo>, </mml:mo><mml:mn>6</mml:mn><mml:mi>g</mml:mi><mml:mo>−</mml:mo><mml:mn>9</mml:mn><mml:mo>)</mml:mo><mml:mo>, </mml:mo><mml:mo>(</mml:mo><mml:mn>13</mml:mn><mml:mo>, </mml:mo><mml:mn>6</mml:mn><mml:mi>g</mml:mi><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives>.</p> </abstract>