A recursive construction of t‐wise uniform permutations. Issue 3 (7th August 2013)

• Record Type: Journal Article
• Authors: Finucane, Hilary; Peled, Ron; Yaari, Yariv
• Is Part Of: Random structures & algorithms. Volume 46:Issue 3(2015)
• Page Start: 531
• Page End: 540
• Publication Date: 2013-08-07
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20509
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We present a recursive construction of a (2<italic>t</italic> + 1)‐wise uniform set of permutations on 2<italic>n</italic> objects using a <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82d3p3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20509:rsa20509-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo><mml:mo>−</mml:mo><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mo>·</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> combinatorial design, a <italic>t</italic>‐wise uniform set of permutations on <italic>n</italic> objects and a (2<italic>t</italic> + 1)‐wise uniform set of permutations on <italic>n</italic> objects. Using the complete design in this procedure gives a <italic>t</italic>‐wise uniform set of permutations on <italic>n</italic> objects whose size is at most <italic>t</italic><sup>2<italic>n</italic></sup>, the first non‐trivial construction of an infinite family of <italic>t</italic>‐wise uniform sets for <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjb82d3r4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20509:rsa20509-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. If a non‐trivial design with suitable parameters is found, it will imply a corresponding improvement in the construction. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 531–540, 2015</p> </abstract>