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Tillich-Zémor code[1]

Alternative Names: Quantum \((n, m, r)\)-structured LDPC code.

Description

A family of \([[n^2 + m^2, (n - \text{rank}([C \mid M]) )^2 + (m - \text{rank}([C \mid M]^\top) )^2, d]]\) quantum LDPC codes constructed via the hypergraph product of two classical \((n, m, r)\)-structured LDPC seed codes

A code’s parity-check matrix \(H = [C \mid M]\) consists of an \(m \times m\) circulant core \(C\) with column weight \(2\) (enabling linear-time encoding) and an \(m \times (n-m)\) matrix \(M\) with column weight \(r \geq 3\) and no zero rows. The resulting code has block length \(N = n^2 + m^2\) and code dimension \(K = (n - \text{rank}([C \mid M]) )^2 + (m - \text{rank}([C \mid M]^\top) )^2\).

Protection

The structured construction inherits the classical seed code’s sub-linear distance scaling of order \(O(n^{\frac{r-2}{r-1}+\epsilon})\) while preserving efficient encoding; see Ref. [2] for distance upper bounds.

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Higher-dimensional homological product code
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
Hypergraph product (HGP) code
Tillich-Zémor code

References

[1]
J. Tillich and G. Zemor, “On the minimum distance of structured LDPC codes with two variable nodes of degree 2 per parity-check equation”, 2006 IEEE International Symposium on Information Theory 1549 (2006) DOI
[2]
F. Arnault, P. Gaborit, W. Rozendaal, N. Saussay, and G. Zémor, “Upper Bounds on the Minimum Distance of Structured LDPC Codes”, (2025) arXiv:2501.19125
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Zoo Code ID: tillichzemor

Cite as:
“Tillich-Zémor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/tillichzemor, arXiv:2606.11484
BibTeX:
@incollection{eczoo_tillichzemor,
title={Tillich-Zémor code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/tillichzemor}
}
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Permanent link:
https://errorcorrectionzoo.org/c/tillichzemor

Cite as:

“Tillich-Zémor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/tillichzemor, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/balanced_product/tensor/singlesector/hypergraph/tillichzemor.yml.