Codes and Modular Forms

The following sections are included:

• Foreword
• Contents
• Introduction

The following sections are included:

• Lattices
  • Sphere Packing Problem
  • Determinant
  • The Theta Series
  • Duality
  • Modular Lattices
• Codes
  • Balls
  • Dimension
  • Weight Enumerators
  • Self-dual Codes
• Dictionary
  • Construction A₄
  • Generalized Constructions A
• Appendix: Modular Forms and Atkin-Lehner Involutions
  • Modular Forms on Γ = SL₂(ℤ)
  • Cusp Form of Weight k
  • Atkin-Lehner Involutions

The following sections are included:

• Broué-Enguehard Map
• Ozeki Map
• Rankin-Cohen Brackets
  • Background
  • RC vs. BE
• Half-integral Weight
  • Background
  • Some Special Theta Series
  • Quaternary Codes
• Higher Levels
  • Background
  • Generalized Construction A
  • Summary of Results
• Appendix: Invariants and Codes
  • Two Variables
  • Three Variables

The following sections are included:

• Introduction
• Genus 2
• Genus 3
• Genus 4
• Higher Genera and Larger Alphabets
  • Weight Enumerators and MacWilliams Identities
  • Invariant Rings
  • Shadows and Weight Enumerators
  • Construction of Siegel Modular Forms
  • Molien Series for Small Cases

The following sections are included:

• Background
• Jacobi Forms and Theta Series
• Jacobi Forms
• Ozeki Polynomials
  • Weight Enumerators
  • Bannai-Ozeki Map
  • Gleason Formulas
• Improvements
  • m = 1
  • m = 2

The following sections are included:

• Introduction
• Codes over the Field of Order 4
  • Notations and Definitions
  • Type II Codes over ${\mathbb{F}}_4$
  • Lattices
  • Constructions of Jacobi Forms over K
  • Construction of Hilbert-Siegel Modular Forms of Genus g over K
• Self-Dual Codes over Polynomial Rings ℤ_2m[x]/〈g(x)〉
  • Notations and Definitions
  • Codes over the Rings ℤ_2m[x]/〈g(x)〉
  • Jacobi Forms over the Totally Real Field K
  • Weight Enumerators and MacWilliams Relations
  • Shadows

The following sections are included:

• Introduction
• Some Preliminaries
  • Shadow of a Type I Lattice
  • Shadow of a Type I Code
  • Generalized Shadow of a Self-dual Code
• Construction of Type I and Type II Codes from Shadows of Type I Codes
• Construction of Self-dual Codes from Generalized Shadows of Self-dual Codes
• Some Examples

The following sections are included:

• Introduction
• Byte Error-control Codes over R and their Byte Weight Enumerators in Genus g
• Jacobi Forms of Genus g
• Lattices and Byte Weight Enumerators in Genus g
• Invariant Space Containing Byte Weight Enumerators in Genus g of Type II Codes over R
• Determination of Jacobi and Siegel Modular Forms of Genus g from Type II Codes over R
  • The case R = ℤ_2m
  • The case R = Σ_2m
• Determination of Jacobi Forms and Siegel Modular Forms of Genus g from Type I Codes over R
• Partial Epstein Zeta Functions and their Functional Equation

The following sections are included:

• Introduction
• Some Preliminaries
  • Byte Weight Enumerators in Genus g
  • Jacobi Form over a Totally Real Field
  • Lattices Induced by Codes over ${\mathbb{F}}_p$
• Theta Series for a Lattice and Byte Weight Enumerator of a Code over ${\mathbb{F}}_p$
• Invariant Spaces and Jacobi Forms over k_p
• Siegel Modular Forms of Genus g over k_p
  • Determination of Siegel Modular Forms over k_p
• Invariant Space and Siegel Modular Forms of Genus g over k_p
• Partial Epstein Zeta Functions and their Functional Equation