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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.

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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 codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Higher-dimensional homological product code
Hypergraph product (HGP) codeGeneralized homological-product Lattice stabilizer QLDPC CSS 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.), NaN. https://errorcorrectionzoo.org/c/tillichzemor
BibTeX:
@incollection{eczoo_tillichzemor, title={Tillich-Zémor code}, booktitle={The Error Correction Zoo}, year={NaN}, editor={Albert, Victor V. and Faist, Philippe}, 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.), NaN. https://errorcorrectionzoo.org/c/tillichzemor

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