# 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}
}
• Published 10 November 2011
• Mathematics
• arXiv: Number Theory
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…
2 Citations
Configurations of Extremal Type II Codes via Harmonic Weight Enumerators
• Mathematics, Computer Science
Journal de Théorie des Nombres de Bordeaux
• 2019
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
• Mathematics, Computer Science
• 2015
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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