On the Classification of Type II Codes of Length 24

@article{Elkies2010OnTC,
  title={On the Classification of Type II Codes of Length 24},
  author={Noam D. Elkies and Scott Duke Kominers},
  journal={SIAM J. Discret. Math.},
  year={2010},
  volume={23},
  pages={2173-2177}
}
We give a new, purely coding-theoretic proof of Koch's criterion on the tetrad systems of Type II codes of length 24 using the theory of harmonic weight enumerators. This approach is inspired by Venkov's approach to the classification of the root systems of Type II lattices in $\mathbb{R}^{24}$ and gives a new instance of the analogy between lattices and codes. 
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3 Citations

3 Citations

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References

SHOWING 1-10 OF 24 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 the Classification and Enumeration of Self-Dual Codes
Self-Dual Codes
A survey of self-dual codes, written for the Handbook of Coding Theory. Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory.…
On the Enumeration of Self-Dual Codes
Lattices and Codes: A Course Partially Based on Lectures by Friedrich Hirzebruch
TLDR
Lattices over Integers of Number Fields and Self-Dual Codes, theta functions and Weight Enumerators, and even Unimodular Lattices are studied.
The Binary Self-Dual Codes of Length up to 32: A Revised Enumeration
The mass of extremal doubly-even self-dual codes of length 40
TLDR
The mass of extremal doubly-even self-dual binary codes of length 40 is determined and it follows that there are at least 12579 such codes.
Good self dual codes exist
Definite quadratische formen der dimension 24 und diskriminante 1
Sphere Packings, Lattices and Groups
The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to…
...
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