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… 
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.

References

SHOWING 1-10 OF 37 REFERENCES
On Harmonic Weight Enumerators of Binary Codes
  • C. Bachoc
  • Computer Science, Mathematics
    Des. Codes Cryptogr.
  • 1999
TLDR
Some new polynomials associated to a linear binary code and a harmonic function of degree k are defined, which classify the extremal even formally self-dual codes of length 12 and can compute some information on the intersection numbers of the code.
On Error-Correcting Codes and Invariant Linear Forms
TLDR
It is proved that the t-designs afforded by the codewords of any fixed weight exhibit extra regularity with respect to $( t + 2 )$-sets.
On the Classification of Type II Codes of Length 24
TLDR
A new, purely coding-theoretic proof of Koch's criterion on the tetrad systems of Type II codes of length 24 is given using the theory of harmonic weight enumerators and gives a new instance of the analogy between lattices and codes.
Hahn Polynomials, Discrete Harmonics, and t-Designs
It is shown that certain Hahn polynomials and their q-analogues play in combinatorics a similar role as Gegenbauer polynomials in real Euclidean geometry. The concept of harmonic function on a fiber
WEIGHT POLYNOMIALS OF SELF-DUAL CODES AND THE MacWILLIAMS IDENTITIES
TLDR
The MacWilliams identities are extended to self-dual codes over larger fields and it is shown that this restriction, for codes over GF(2) and GF(3), is that the weight polynomial must lie in an explicitly described free polynometric ring.
A theorem on the distribution of weights in a systematic code
TLDR
The spectrum of a systematic code determines uniquely the spectrum of its dual code (the orthogonal vector space) and the two sets of integers are related by a system of linear equations.
An Upper Bound for Self-Dual Codes
A mass formula for unimodular lattices with no roots
TLDR
A mass formula for n-dimensional unimodular lattices having any prescribed root system is derived using Katsurada's formula for the Fourier coefficients of Siegel Eisenstein series and better lower bounds are computed on the number of inequivalent unimodULAR lattices in dimensions 26 to 30 than those afforded by the Minkowski-Siegel mass constants.
...
...