4D and 5D Layer Codes through Color Routing
Abstract
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(nD/(D−2)), k, Θ(dn1/(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.
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BibTeX
@misc{yuan2026layer,
title = {{4D and 5D Layer Codes through Color Routing}},
author = {Andrew C. Yuan and Nouédyn Baspin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.18961},
eprint = {2605.18961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.18961},
}
RIS
TY - UNPB
TI - 4D and 5D Layer Codes through Color Routing
AU - Andrew C. Yuan
AU - Nouédyn Baspin
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2605.18961
UR - https://arxiv.org/abs/2605.18961
AB - 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.
ER -