Weighted Generating Functions for Type II Lattices and Codes

@article{Elkies2013WeightedGF,
  title={Weighted Generating Functions for Type II Lattices and Codes},
  author={Noam D. Elkies and Scott Duke Kominers},
  journal={arXiv: Number Theory},
  year={2013},
  pages={63-108}
}
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… 
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2 Citations

2 Citations

Configurations of Extremal Type II Codes via Harmonic Weight Enumerators
TLDR
It is shown that for n ∈ {8, 24, 32, 48, 56, 72, 96} every extremal Type II code of length n is generated by its codewords of minimal weight.
Configurations of Extremal Type II Codes
TLDR
Every extremal Type II code of length n is generated by its codewords of minimal weight, and "$t\frac12$-designs" is introduced as a discrete analog of Venkov's spherical designs of the same name.

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