Cyclic Hypergraph Product Code[1]
Description
The hypergraph product code constructed using two low-weight circulant matrices. The code family \(\mathrm{C2}\) is the product of a cyclic LDPC code with itself, and the family \(\mathrm{CxR}\) is the product of the cyclic code with a repetition code.
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 \(a_i\in\{0,1\}\).
Decoding
BP-OSD decoderCousins
- 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 \([[2nd,2k,d]]\).
- La-cross code— The La-cross code is a reduced block length, full-rank cyclic HGP code with generator polynomials of the form \(1+x+x^k\)
- Generalized bicycle (GB) code— Cyclic HGP codes and GB codes both use circulant matrices as building blocks.
Primary Hierarchy
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Homological product codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Hypergraph product (HGP) codeCSS QLDPC Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based Qubit QECC Quantum
Parents
A cyclic hypergraph product code is a hypergraph product code constructed using two circulant matrices.
Abelian two-block group-algebra codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
A cyclic HGP 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 [3][2; 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 [4; Exam. 6]. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family [4; Exam. 5].
References
- [1]
- A. Aydin, N. Delfosse, and E. Tham, “Cyclic Hypergraph Product Code”, (2026) arXiv:2511.09683
- [2]
- H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400
- [3]
- F. A. Mian, O. Gwilliam, and S. Krastanov, “Multivariate Multicycle Codes for Complete Single-Shot Decoding”, (2026) arXiv:2601.18879
- [4]
- 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 (2026-07-18) — most recent
- 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