Pinwheel code[1]
Description
A geometrically local binary LDPC code defined on planar graphs obtained from the pinwheel tiling [2]. Both bits and checks live on vertices of the graph. If \(L_N\) is the graph Laplacian at generation \(N\), the undepleted check matrix is \(\tilde H_N=(L_N-\mathbb{I})\bmod 2\), and the actual parity-check matrix \(H_N\) is obtained by removing an evenly spaced fraction of boundary checks.
The construction was introduced as a concrete aperiodic seed code whose hypergraph products realize local Type-I and Type-II fracton models.
Protection
For boundary-depletion period \(p\), the family has \(k\approx \sqrt{n}/p\) and distance \(d\sim n\) [1], saturating the classical BPT bound \(k\sqrt{d}=O(n)\). The same work gives numerical evidence for confinement.Rate
The encoding rate vanishes asymptotically because \(k\sim\sqrt{n}\) while the block length grows with the substitution generation [1].Cousins
- Laplacian code— The pinwheel code is derived from the graph Laplacian of the pinwheel tiling, with a fraction of boundary checks removed.
- Hypergraph product (HGP) code— The hypergraph product of a pinwheel code with a cyclic repetition code yields a local Type-I fracton model in three dimensions, while the hypergraph product of two pinwheel codes yields a local Type-II fracton model in four dimensions [1].
- Repetition code— The hypergraph product of a pinwheel code with a cyclic repetition code yields a local Type-I fracton model in three dimensions [1].
- Fracton stabilizer code— The hypergraph product of a pinwheel code with a cyclic repetition code yields a local Type-I fracton model in three dimensions, while the hypergraph product of two pinwheel codes yields a local Type-II fracton model in four dimensions [1].
Primary Hierarchy
References
- [1]
- Y. Tan, B. Roberts, N. Tantivasadakarn, B. Yoshida, and N. Y. Yao, “Fracton models from product codes”, Physical Review Research 7, (2025) arXiv:2312.08462 DOI
- [2]
- J. H. Conway and C. Radin, “Quaquaversal tilings and rotations”, Inventiones Mathematicae 132, 179 (1998) DOI
Page edit log
- Victor V. Albert (2026-04-22) — most recent
Cite as:
“Pinwheel code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/pinwheel