# Hemicubic code[1]

## Description

Homological code constructed out of cubes in high dimensions. The hemicubic code family has asymptotically diminishing soundness that scales as order \(\Omega(1/\log n)\), locality of stabilizer generators scaling as order \(O(\log n)\), and distance \(\Theta(\sqrt{n})\).

## Parent

## Cousins

- Quantum locally testable code (QLTC) — The hemicubic code family has asymptotically diminishing soundness that scales as order \(\Omega(1/\log n)\), locality of stabilizer generators scaling as order \(O(\log n)\), and distance \(\Theta(\sqrt{n})\). Soundness amplification and AEL distance amplification can also yield improvements in various parameters [2; Table 3]. Application of generalized distance balancing [3] to hemicubic codes using an asymptotically good classical code of length \(t\) yields \(O(1/(\log(n) t^2))\) soundness and \(\Theta(\sqrt{n}t)\) distance while maintaining locality scaling and at the expense of a dimension scaling as \(\Theta(t^2)\) [4].
- Distance-balanced code — Application of generalized distance balancing [3] to hemicubic codes using an asymptotically good classical code of length \(t\) yields \(O(1/(\log(n) t^2))\) soundness and \(\Theta(\sqrt{n}t)\) distance while maintaining locality scaling and at the expense of a dimension scaling as \(\Theta(t^2)\) [4].
- Hypercube code

## References

- [1]
- A. Leverrier, V. Londe, and G. Zémor, “Towards local testability for quantum coding”, Quantum 6, 661 (2022) arXiv:1911.03069 DOI
- [2]
- A. Wills, T.-C. Lin, and M.-H. Hsieh, “Tradeoff Constructions for Quantum Locally Testable Codes”, (2024) arXiv:2309.05541
- [3]
- S. Evra, T. Kaufman, and G. Zémor, “Decodable quantum LDPC codes beyond the \(\sqrt{n}\) distance barrier using high dimensional expanders”, (2020) arXiv:2004.07935
- [4]
- A. Wills, T.-C. Lin, and M.-H. Hsieh, “General Distance Balancing for Quantum Locally Testable Codes”, (2023) arXiv:2305.00689

## Page edit log

- Victor V. Albert (2022-09-26) — most recent

## Cite as:

“Hemicubic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hemicubic