The complex conjugate invariants of Clifford groups

TL;DR

This paper investigates the polynomial invariants of the complex Clifford group acting on conjugate polynomials, establishing that certain orbit properties imply higher design strength, and confirms a conjecture about these invariants.

Contribution

It extends known results on Clifford group invariants to the complex case, proving a conjecture relating projective 4-designs and 5-designs for group orbits.

Findings

The dimension of the invariant space for (N_1,N_2)=(5,5) is 2.

Orbit of the complex Clifford group as a projective 4-design implies it is a 5-design.

Confirmed the conjecture by Zhu et al. regarding invariants and design properties.

Abstract

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.