Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The MacWilliams Identity for Krawtchouk Association Schemes

Izzy Friedlander

Abstract

The MacWilliams Identity is a well established theorem relating the weight enumerator of a code to the weight enumerator of its dual. The ability to use a known weight enumerator to generate the weight enumerator of another through a simple transform proved highly effective and efficient. An equivalent relation was also developed by Delsarte which linked the eigenvalues of any association scheme to the eigenvalues of it's dual association scheme but this was less practical to use in reality. A functional transform was developed for some specific association schemes including those based on the rank metric, the skew rank metric and Hermitian matrices. In this paper those results are unified into a single consistent theory applied to these "Krawtchouk association schemes" using a $b$-algebra. The derivatives formed using the $b$-algebra have also been applied to derive the moments of the weight distribution for any code within these association schemes.

The MacWilliams Identity for Krawtchouk Association Schemes

Abstract

The MacWilliams Identity is a well established theorem relating the weight enumerator of a code to the weight enumerator of its dual. The ability to use a known weight enumerator to generate the weight enumerator of another through a simple transform proved highly effective and efficient. An equivalent relation was also developed by Delsarte which linked the eigenvalues of any association scheme to the eigenvalues of it's dual association scheme but this was less practical to use in reality. A functional transform was developed for some specific association schemes including those based on the rank metric, the skew rank metric and Hermitian matrices. In this paper those results are unified into a single consistent theory applied to these "Krawtchouk association schemes" using a -algebra. The derivatives formed using the -algebra have also been applied to derive the moments of the weight distribution for any code within these association schemes.
Paper Structure (26 sections, 29 theorems, 179 equations, 1 figure, 4 tables)

This paper contains 26 sections, 29 theorems, 179 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Association Schemes and Distance Regular Graphs
  3. Preliminaries
  4. The Bose-Mesner Algebra
  5. Metric Association Schemes and Distance Regular Graphs
  6. Generalised Krawtchouk Polynomials
  7. Krawtchouk Association Schemes
  8. Recurrence relation
  9. The b-Krawtchouk Polynomials
  10. Weight Functions
  11. The b-Algebra
  12. The b-Product, b-Power and b-Transform
  13. Fundamental Polynomials
  14. The Generalised MacWilliams Identity
  15. Generalised MacWilliams Identity
  16. ...and 11 more sections

Key Result

Lemma 2.4

Using the properties of an $(\mathscr{X},R)$ symmetric association scheme BrouwerDRG we have that, Conversely, any set of $\{0,1\}$ matrices, $D_0,\ldots,D_n$, that satisfies equation:adjacencyidentity - equation:assocationvi1s is the collection of adjacency matrices of an association scheme.

Figures (1)

  • Figure 2.1: Visualisation of points and relations in an association scheme.

Theorems & Definitions (76)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: TheoryofError
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7: BrouwerDRG
  • Definition 2.8: TheoryofError
  • Theorem 2.9: delsarte1973algebraic
  • ...and 66 more