Cyclic hypergraph product code[1]
Description
Hypergraph product code constructed from two circulant matrices, typically of low weight. The \(\mathrm{C2}\) subfamily is the product of a cyclic LDPC code with itself, while the \(\mathrm{CxR}\) subfamily is the product of the cyclic code with a repetition code [1].
The construction of \(\mathrm{C2}\) uses a single generating polynomial \(\sum a_ix^i\) for both factors, while the construction of \(\mathrm{CxR}\) uses the generating polynomial \(\sum a_ix^i\) along with the polynomial \(1+x\), where \(a_i\in\{0,1\}\).
Protection
A CxC code built from circulant matrices \(A\) and \(B\) of sizes \(a\) and \(b\) and ranks \(r_a\) and \(r_b\) has parameters \([[2ab,2(a-r_a)(b-r_b),\min(d_A,d_B)]]\), where \(d_A\) and \(d_B\) are the distances of the two classical codes [1]. Such codes satisfy balance criteria: they have equal numbers of \(X\)- and \(Z\)-type stabilizer generators, \(\text{rank}(H_X)=\text{rank}(H_Z)=(n-k)/2\), and all generators have the same weight \(w(A)+w(B)\). This yields near-identical performance in the \(X\) and \(Z\) logical bases, which is not guaranteed for a general HGP code [1].Decoding
BP-OSD decoder [1].Cousins
- Quasi-cyclic LDPC (QC-LDPC) code— A classical cyclic LDPC code with parameters \([n,k,d]\) yields a \(\mathrm{C2}\) code with parameters \([[2n^2,2k^2,d]]\) and a \(\mathrm{CxR}\) code with parameters \([[2nd,2k,d]]\) [1].
- La-cross code— Periodic-boundary La-cross codes are CxC codes built from the seed polynomial \(1+x+x^k\); the open-boundary La-cross family instead uses a rectangular full-rank seed [2].
- Generalized bicycle (GB) code— CxC codes and GB codes both use circulant matrices as building blocks [1].
Primary Hierarchy
Generalized homological-product qubit CSS codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
Homological product codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
Hypergraph product (HGP) codeCSS QLDPC Qubit Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Parents
A CxC code is a hypergraph product code constructed using two circulant matrices [1].
Abelian two-block group-algebra codeCSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
A CxC code is an Abelian 2BGA code over \(\mathbb{Z}_{n_1}\times\mathbb{Z}_{n_2}\) whose two group-algebra elements are polynomials in disjoint variables, \(a=a(x)\) and \(b=b(y)\); trivially intersecting supports turn a two-block code into a hypergraph product [4][3; Statements 8 and 12].
Cyclic hypergraph product code
Children
LRESCs are constructed using a hypergraph product of a concatenated LDPC-repetition code with itself.
The toric code can be obtained from a hypergraph product of two repetition codes [5; Exam. 6]. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family [5; Exam. 5].
References
- [1]
- A. Aydin, N. Delfosse, and E. Tham, “Cyclic Hypergraph Product Code”, (2026) arXiv:2511.09683
- [2]
- L. Pecorari, S. Jandura, G. K. Brennen, and G. Pupillo, “High-rate quantum LDPC codes for long-range-connected neutral atom registers”, Nature Communications 16, (2025) arXiv:2404.13010 DOI
- [3]
- H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400
- [4]
- F. A. Mian, O. Gwilliam, and S. Krastanov, “Multivariate Multicycle Codes for Complete Single-Shot Decoding”, (2026) arXiv:2601.18879
- [5]
- A. A. Kovalev and L. P. Pryadko, “Improved quantum hypergraph-product LDPC codes”, 2012 IEEE International Symposium on Information Theory Proceedings 348 (2012) arXiv:1202.0928 DOI
Page edit log
- Victor V. Albert (2026-08-24) — most recent
- Victor V. Albert (2026-08-22)
- Victor V. Albert (2026-08-19)
- Victor V. Albert (2026-07-18)
- Victor V. Albert (2026-06-08)
- Arda Aydin (2026-06-08)
- Victor V. Albert (2026-06-08)
Cite as:
“Cyclic hypergraph product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/cyclic_hgp, arXiv:2606.11484