Abstract
Nebe, Rains and Sloane studied the polynomial invariants for real and complex Clifford groups and they give a new conceptual proof that these invariants are spanned by the set of complete weight enumerators of certain self-dual codes, which was first proved by Runge using modular forms. The purpose of this paper is to show that very similar results can be obtained for the invariants of the complex Clifford group \(\mathcal {X}_m\) acting on the space of conjugate polynomials in \(2^m\) variables of degree \(N_1\) in \(x_f\) and of degree \(N_2\) in their complex conjugates \(\overline{x_f}\). In particular, we show that the dimension of this space is 2, for \((N_1,N_2)=(5,5)\). This solves affirmatively Conjecture 2 given by Zhu, Kueng, Grassl and Gross. In other words if an orbit of the complex Clifford group is a projective 4-design, then it is automatically a projective 5-design.
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Acknowledgements
We thank Nebe, Rains and Sloane for fruitful discussions, especially for rectifying our understanding of the weight enumerator conjecture. The first author thanks TGMRC (Three Gorges Mathematical Research Center) in China Three Gorges University, in Yichang, Hubei, China, for supporting his visits there in April and August 2019 to work on the topics related to this research. The second author is supported by JSPS KAKENHI (17K05164). The third author is supported in part by NSFC (11671258).
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Bannai, E., Oura, M. & Zhao, D. The complex conjugate invariants of Clifford groups. Des. Codes Cryptogr. 89, 341–350 (2021). https://doi.org/10.1007/s10623-020-00819-7
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DOI: https://doi.org/10.1007/s10623-020-00819-7