We introduce and explicit Calderbank-Shor-Steane (CSS) code construction that generalizes the Layer codes to $D=4,5$ dimensions. Much like its predecessor, the present construction is based on embedding quantum low-density parity check (qLDPC) codes; from an $[[n,k,d]]$ code with energy barrier $Δ$, we obtain a $D=4,5$ dimensional Layer code with parameters $[[Θ(n^{D/(D-2)}), k, Θ(dn^{1/(D-2)})]]$ and energy barrier $Ω(Δ)$. Using good qLDPC codes as input, our construction saturates the $D=4,5$ dimensional BPT bounds exactly. The higher dimensional Layer Codes are modular, and thus well suited to architectures composed of modular network patches, despite our physical limitation to three dimensions. We overcome the hurdles encountered by previous generalization attempts through the use of \textit{color routing}, allowing us to resolve the structure of the check layers and line defects.

翻译：我们提出了一种明确的Calderbank-Shor-Steane（CSS）码构造方法，将层码推广至$D=4,5$维。与先前工作类似，本构造基于量子低密度奇偶校验（qLDPC）码的嵌入：给定一个具有能垒$\Delta$的$[[n,k,d]]$码，我们得到参数为$[[\Theta(n^{D/(D-2)}), k, \Theta(dn^{1/(D-2)})]]$且能垒为$\Omega(\Delta)$的$D=4,5$维层码。采用优质qLDPC码作为输入时，我们的构造精确饱和了$D=4,5$维的BPT界。高维层码具有模块化特性，尽管物理空间受限为三维，仍适用于由模块化网络补丁构成的架构。通过引入\textit{彩色路由}，我们克服了此前推广尝试中遇到的障碍，从而成功解析了校验层与线缺陷的结构。