Weighted Generating Functions for Type II Lattices and Codes

Chapter
Part of the Developments in Mathematics book series (DEVM, volume 31)

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 \(\mathfrak{s}\mathfrak{l}_{2}\) 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 

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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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