# NTRU-GKP code[1]

## Description

Multi-mode GKP code whose underlying lattice is utilized in variations of the NTRU cryptosystem [2]. Randomized constructions yield constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability.

The integer-valued \(q\)-symplectic Gram matrix for an \(n\)-mode \(k\)-qubit good NTRU-GKP code is \begin{align} A = \sqrt{\frac{2}{q}}\begin{pmatrix}I & Q\\ 0 & qI \end{pmatrix}~, \tag*{(1)}\end{align} where \(Q\) is a circulant matrix constructed from coefficients of a cyclic polynomial used in the NTRU cryptosystem, and \(I\) is the \(n\)-dimensional identity matrix [1; Prop. 2].

## Rate

## Decoding

## Code Capacity Threshold

## Realizations

## Parents

## Cousin

- Random quantum code — Several NTRU lattices come from randomized constructions, yielding constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability.

## References

- [1]
- J. Conrad, J. Eisert, and J.-P. Seifert, “Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem”, Quantum 8, 1398 (2024) arXiv:2303.02432 DOI
- [2]
- J. Hoffstein, J. Pipher, and J. H. Silverman, “NTRU: A ring-based public key cryptosystem”, Lecture Notes in Computer Science 267 (1998) DOI
- [3]
- L. Babai, “On Lovász’ lattice reduction and the nearest lattice point problem”, Combinatorica 6, 1 (1986) DOI

## Page edit log

- Victor V. Albert (2023-03-27) — most recent

## Cite as:

“NTRU-GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ntru_gkp