Q‐Analogs of Packing Designs. Issue 7 (15th October 2013)

• Record Type: Journal Article
• Authors: Braun, Michael; Reichelt, Jan
• Is Part Of: Journal of combinatorial designs. Volume 22:Issue 7(2014:Jul.)
• Page Start: 306
• Page End: 321
• Publication Date: 2013-10-15
• Subjects: Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21376
• Languages: English
• ISSNs: 1063-8539

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>A <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg5cg04k9c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21376:jcd21376-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>, </mml:mo><mml:mi>k</mml:mi><mml:mo>, </mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives><italic>q</italic>‐packing design is a selection of <italic>k</italic>‐dimensional subspaces of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg5cg04k8t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21376:jcd21376-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi mathvariant="double-struck">F</mml:mi><mml:mi>q</mml:mi><mml:mi>n</mml:mi></mml:msubsup></mml:math></alternatives> such that each <italic>t</italic>‐dimensional subspace is contained in at most one element of the collection. A successful approach adopted from the Kramer–Mesner method of prescribing a group of automorphisms was applied by Kohnert and Kurz to construct some constant dimension codes with moderate parameters that arise by <italic>q</italic>‐packing designs. In this paper, we recall this approach and give a version of the Kramer–Mesner method breaking the condition that the whole <italic>q</italic>‐packing design must admit the prescribed group of automorphisms. Afterwards, we describe the basic idea of an algorithm to tackle the integer linear optimization problems representing the <italic>q</italic>‐packing design construction by means of a metaheuristic approach. Finally, we give some improvements on the size of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg5cg04kcg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21376:jcd21376-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>, </mml:mo><mml:mn>3</mml:mn><mml:mo>, </mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives><italic>q</italic>‐packing designs.</p> </abstract>