论文标题
克利福德小组的复杂共轭不变
The complex conjugate invariants of Clifford groups
论文作者
论文摘要
Nebe，Rains和Sloane研究了真实和复杂的Clifford组的多项式不变性，它们将不变性与某些自我偶发代码的完整重量枚举者的空间联系起来。本文的目的是表明，对于复杂的Clifford组$ \ Mathcal的不变性，可以获得非常相似的结果，该{X} _M $在$ 2^m $ $ $ $ n_1 $ n_1 $ n_1 in $ x_f $的$ 2^m $变量的$ 2^m $变量中作用于$ x_f $的$ x_f $，并在其复杂的conjugugatate $ n_2 $中}特别是，我们表明该空间的尺寸为$ 2 $，对于$（N_1，N_2）=（5,5）$。这解决了在朱，库恩，格拉斯和总体上给出的猜想2。换句话说，如果Complex Clifford组的轨道是投影$ 4 $ -DESIGN，那么它实际上是一个投影$ 5 $ -DESIGN。
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.
