Clifford群的复共轭的不变量
The complex conjugate invariants of Clifford groups
摘要：
Nebe，Rains和Sloane研究了实Clifford群和复Clifford群的多项式不变量，他们将不变量与某些自对偶码的完全权枚举空间联系起来。本文的目的是表明，非常相似的结果可以得到的不变量的复Clifford群$\mathcal{X}_m$的共轭多项式的空间上的$2^m$变量的次数$N_1$在$x_f$和次数$N_2$在其复共轭$\overline{x_f}$。特别是，我们表明，这个空间的尺寸是$2$，为$(N_1,N_2)=(5,5)$。这肯定地解决了Zhu，Kueng，Grassl和Gross的猜想2。换句话说，如果复Clifford群的轨道是一个射影$4$ -设计，那么它自动是一个射影$5$ -设计。
英文摘要：
Nebe, Rains and Sloane studied the polynomial invariants for real and complex Clifford groups and they relate the invariants to the space of complete weight enumerators of certain self-dual codes. 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 the Conjecture 2 given in Zhu, Kueng, Grassl and Gross affirmatively. 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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