Skip to Main Content
HBS Home
  • About
  • Academic Programs
  • Alumni
  • Faculty & Research
  • Baker Library
  • Giving
  • Harvard Business Review
  • Initiatives
  • News
  • Recruit
  • Map / Directions
Faculty & Research
  • Faculty
  • Research
  • Featured Topics
  • Academic Units
  • …→
  • Harvard Business School→
  • Faculty & Research→
Publications
Publications
  • 2013
  • Chapter
  • Quadratic and Higher Degree Forms

Weighted Generating Functions for Type II Lattices and Codes

By: Noam D. Elkies and Scott Duke Kominers
  • Format:Print
ShareBar

Abstract

We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of Euclidean lattices. Namely, we use the finite-dimensional representation theory of sl2sl2 to derive a decomposition theorem for the spaces of discrete homogeneous polynomials in terms of the spaces of discrete harmonic polynomials, and prove a generalized MacWilliams identity for harmonic weight enumerators. We then present several applications of harmonic weight enumerators, corresponding to some uses of weighted theta functions: an equivalent characterization of t-designs, the Assmus–Mattson Theorem in the case of extremal Type II codes, and configuration results for extremal Type II codes of lengths 8, 24, 32, 48, 56, 72, and 96.

Keywords

Mathematical Methods

Citation

Elkies, Noam D., and Scott Duke Kominers. "Weighted Generating Functions for Type II Lattices and Codes." In Quadratic and Higher Degree Forms. Vol. 31, edited by Alladi Krishnaswami, Manjul Bhargava, David Savitt, and Pham Huu Tiep, 63–108. Developments in Mathematics. Springer, 2013.
  • Find it at Harvard
  • Read Now

About The Author

Scott Duke Kominers

Entrepreneurial Management
→More Publications

More from the Authors

    • June 2025
    • Journal of Finance

    Collusion in Brokered Markets

    By: John William Hatfield, Scott Duke Kominers and Richard Lowery
    • March 14, 2025
    • Harvard Crimson

    Harvard Students Should Ignore Calls to Boycott Israel Trek

    By: Jesse M. Fried, Paul A. Gompers, Scott Kominers and Mark C. Poznansky
    • March 2025
    • Faculty Research

    O2X: Optimizing to the X

    By: Scott Duke Kominers, Thomas Jennings and Maisie Wiltshire-Gordon
More from the Authors
  • Collusion in Brokered Markets By: John William Hatfield, Scott Duke Kominers and Richard Lowery
  • Harvard Students Should Ignore Calls to Boycott Israel Trek By: Jesse M. Fried, Paul A. Gompers, Scott Kominers and Mark C. Poznansky
  • O2X: Optimizing to the X By: Scott Duke Kominers, Thomas Jennings and Maisie Wiltshire-Gordon
ǁ
Campus Map
Harvard Business School
Soldiers Field
Boston, MA 02163
→Map & Directions
→More Contact Information
  • Make a Gift
  • Site Map
  • Jobs
  • Harvard University
  • Trademarks
  • Policies
  • Accessibility
  • Digital Accessibility
Copyright © President & Fellows of Harvard College.