Weighted Generating Functions for Type II Lattices and Codes

  • Noam D. ElkiesEmail author
  • Scott Duke Kominers
Part of the Developments in Mathematics book series (DEVM, volume 31)


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 \(\mathfrak{s}\mathfrak{l}_{2}\) 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.

Key words and Phrases

Harmonic polynomial Weight enumerator Binary code Extremal code Theta function Lattice Design Configuration result 

Mathematics Subject Classification (2010):

Primary: 94B05 Secondary: 05B05 11H71 33C50 33C55 



The authors thank Zachary Abel, Henry Cohn, John H. Conway, John F. Duncan, Benedict H. Gross, Abhinav Kumar, Barry Mazur, Gabriele Nebe, Ken Ono, Vera Pless, Eric M. Rains, Shrenik Shah, and the anonymous referee for helpful comments and suggestions. During parts of this research, Elkies was supported by NSF grants DMS-0501029 and DMS-1100511, and Kominers was supported by the Harvard College Program for Research in Science and Engineering (PRISE), a Harvard Mathematics Department Highbridge Fellowship, an NSF Graduate Research Fellowship, a Yahoo! Key Scientific Challenges Program Fellowship, a Terence M. Considine Fellowship funded by the John M. Olin Center, and an AMS-Simons Travel Grant.This work includes a part of the second author ’s undergraduate thesis [Kom09b].


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Society of Fellows, Harvard University, and Becker Friedman Institute for Research in EconomicsUniversity of ChicagoCambridgeUSA

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