We construct good GKP (Gottesman-Kitaev-Preskill) codes (in the sense of Conrad, Eisert and Seifert proposed) from standard short integer solution lattices (SIS) as well as from ring SIS and module SIS lattices, R-SIS and M-SIS lattices, respectively. These lattice are crucial for lattice-based cryptography. Our construction yields GKP codes with distance $\sqrt{n/πe}$. This compares favorably with the NTRU-based construction by Conrad et al. that achieves distance $Ω(\sqrt{n/q}),$ with $n\le q^2/0.28$. Unlike their codes, our codes do not have secret keys that can be used to speed-up the decoding. However, we present a simple decoding algorithm that, for many parameter choices, experimentally yields decoding results similar to the ones for NTRU-based codes. Using the R-SIS and M-SIS construction, our simple decoding algorithm runs in nearly linear time. Following Conrad, Eisert and Seifert's work, our construction of GKP codes follows directly from an explicit, randomized construction of symplectic lattices with (up to constants $\approx 1$) minimal distance $(1/σ_{2n})^{1/2n}\approx \sqrt{\frac{n}{πe}}$, where $σ_{2n}$ is the volume of the 2n-dimensional unit ball. Before this result, Buser and Sarnak gave a non-constructive proof for the existence of such symplectic lattices.

翻译：我们利用标准短整数解（SIS）晶格，以及环SIS（R-SIS）和模SIS（M-SIS）晶格，构造了优良的GKP（Gottesman-Kitaev-Preskill）码（符合Conrad、Eisert和Seifert提出的标准）。这些晶格对基于晶格的密码学至关重要。我们的构造产生了距离为$\sqrt{n/πe}$的GKP码。这与Conrad等人基于NTRU的构造（其距离为$Ω(\sqrt{n/q})$，其中$n\le q^2/0.28$）相比具有优势。不同于他们的码，我们的码没有可用于加速解码的密钥。然而，我们提出了一种简单的解码算法，对于许多参数选择，实验表明其解码结果与基于NTRU的码相似。利用R-SIS和M-SIS构造，我们的简单解码算法运行时间接近线性。继Conrad、Eisert和Seifert的工作之后，我们直接通过对辛普理克晶格进行显式随机化构造（在常数$\approx 1$以内）得到了最小距离$(1/σ_{2n})^{1/2n}\approx \sqrt{\frac{n}{πe}}$，其中$σ_{2n}$是$2n$维单位球的体积。在此结果之前，Buser和Sarnak为这类辛普理克晶格的存在性给出了非构造性证明。