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J. Korean Math. Soc. 2013; 50(3): 509-527

Printed May 1, 2013

https://doi.org/10.4134/JKMS.2013.50.3.509

Copyright © The Korean Mathematical Society.

Explicit expression of the Krawtchouk polynomial via a discrete Green's function

Gil Chun Kim and Yoonjin Lee

Ewha Womans University, Ewha Womans University

Abstract

A Krawtchouk polynomial is introduced as the classical Mac-Williams identity, which can be expressed in weight-enumerator-free form of a linear code and its dual code over a Hamming scheme. In this paper we find a new explicit expression for the $p$-number and the $q$-number, which are more generalized notions of the Krawtchouk polynomial in the $P$-polynomial schemes by using an extended version of a discrete Green's function. As corollaries, we obtain a new expression of the Krawtchouk polynomial over the Hamming scheme and the Eberlein polynomial over the Johnson scheme. Furthermore, we find another version of the MacWilliams identity over a Hamming scheme.

Keywords: discrete Green's function, $P$-polynomial scheme, $p$-number, $q$-number, Krawtchouk polynomial, Eberlein polynomial

MSC numbers: Primary 58B34, 58J42, 81T75

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