Abstract
In a previous paper we showed that if one particular Fourier coefficient of the Siegel theta series of degree 4 for a 32-dimensional even unimodular extremal lattice is known then the other Fourier coefficients of the series are in principle determined. In this paper we choose the quaternary positive definite symmetric matrix \(\mathfrak {T}_{40}\), and calculate the Fourier coefficient \(a(\mathfrak {T}_{40},\mathcal{L}_{32})\) of the Siegel theta series of degree 4 associated with the five even unimodular extremal lattices which come from the five binary self-dual extremal [32,16,8] codes. As a result we can show that the five Siegel theta series of degree 4 associated with the five 32-dimensional even unimodular extremal lattices are distinct.
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Acknowledgments
The authors would like to express their thanks to the referee of the present article for giving us very useful comments, which greatly improved the paper. Also they would like to express their cordial thanks to Dr. N.J.A. Sloane who read their manuscript very carefully and suggested them many practical improvements for the smooth writing of their paper. This work was supported by JSPS KAKENHI Grant Number 25400014.
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Communicated by Jens Funke.
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Oura, M., Ozeki, M. Distinguishing Siegel theta series of degree 4 for the 32-dimensional even unimodular extremal lattices. Abh. Math. Semin. Univ. Hambg. 86, 19–53 (2016). https://doi.org/10.1007/s12188-016-0120-y
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DOI: https://doi.org/10.1007/s12188-016-0120-y