The good GKP codes conjecture for random NTRU lattices

Let $L$ be the NTRU lattice specified by the basis in equation (HCS), and let

$$R_q=\mathbb{Z}_q[x]/\langle x^n-1\rangle$$

with $h$ selected at random from $R_q$. For a GKP code with

$$\mathcal{C}=\sqrt{\frac{\lambda}{q}}L,$$

Good GKP codes conjecture. The code is likely to be good, encodes $k=n$ qubits, and has distance

$$\Delta\geq\min\left\{\sqrt{\frac{n}{\lambda\pi e}},\sqrt{\frac{q}{\lambda}}\right\}.$$

This conjecture is motivated by numerical evidence for random NTRU lattices and by known lower bounds on their shortest-vector lengths; it predicts that suitable random NTRU-lattice constructions yield good GKP codes.