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On Krawtchouk Transforms

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Book cover Intelligent Computer Mathematics (CICM 2010)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6167))

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Abstract

Krawtchouk polynomials appear in a variety of contexts, most notably as orthogonal polynomials and in coding theory via the Krawtchouk transform. We present an operator calculus formulation of the Krawtchouk transform that is suitable for computer implementation. A positivity result for the Krawtchouk transform is shown. Then our approach is compared with the use of the Krawtchouk transform in coding theory where it appears in MacWilliams’ and Delsarte’s theorems on weight enumerators. We conclude with a construction of Krawtchouk polynomials in an arbitrary finite number of variables, orthogonal with respect to the multinomial distribution.

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Author information

Authors and Affiliations

  1. Southern Illinois University, Carbondale, IL, 62901, U.S.A.

    Philip Feinsilver

  2. IECN and LORIA, Nancy-Université, Université Henri Poincaré, 54506, Vandoeuvre-lès-Nancy, France

    René Schott

Authors
  1. Philip Feinsilver
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  2. René Schott
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Editor information

Editors and Affiliations

  1. German Research Center for Artificial Intelligence Bremen, Germany

    Serge Autexier

  2. Karlsruhe Institute of Technology, Germany

    Jacques Calmet

  3. CEDRIC/CNAM, Paris, France

    David Delahaye

  4. American Mathematical Society, Ann Arbor, USA

    Patrick D. F. Ion

  5. INRIA Sophia-Antipolis, France

    Laurence Rideau

  6. École Nationale Supérieure d’Informatique pour l’industrie et l’Entreprise, Evry, France

    Renaud Rioboo

  7. School of Computer Science, Edgbaston, University of Birmingham, B15 2TT, Birmingham, UK

    Alan P. Sexton

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© 2010 Springer-Verlag Berlin Heidelberg

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Feinsilver, P., Schott, R. (2010). On Krawtchouk Transforms. In: Autexier, S., et al. Intelligent Computer Mathematics. CICM 2010. Lecture Notes in Computer Science(), vol 6167. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14128-7_7

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  • DOI: https://doi.org/10.1007/978-3-642-14128-7_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14127-0

  • Online ISBN: 978-3-642-14128-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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