Parity-check tensor-product code

Parity-check tensor-product code[1]

Description

A matrix-based code constructed out of two linear binary or \(q\)-ary codes with parity-check matrices \(H_A,H_B\) such its parity-check matrix is \(H_A \otimes H_B\). Its dual has parity-check matrix \(G_A\otimes G_B\), where \(G_{A,B}\) are the generator matrices of the two underlying codes [2].

Notes

See Ref. [3] for an exposition.

Cousin

Tensor-product code— Tensor-product codewords (parity-check tensor-product parity-check matrices) are constructed via an outer product of the underlying codes (parity-check matrices).

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

\(q\)-ary codeECC

Additive \(q\)-ary codeECC

Linear \(q\)-ary codeECC

Parents

Linear \(q\)-ary code

Parity-check tensor-product code

References

[1]: J. Wolf, “On codes derivable from the tensor product of check matrices”, IEEE Transactions on Information Theory 11, 281 (1965) DOI
[2]: D. Ostrev, D. Orsucci, F. Lázaro, and B. Matuz, “Classical product code constructions for quantum Calderbank-Shor-Steane codes”, Quantum 8, 1420 (2024) arXiv:2209.13474 DOI
[3]: J. K. Wolf, “An Introduction to Tensor Product Codes and Applications to Digital Storage Systems”, 2006 IEEE Information Theory Workshop - ITW ’06 Chengdu (2006) DOI

Page edit log

Victor V. Albert (2025-02-20) — most recent

Cite as:

“Parity-check tensor-product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/parity_check_tensor

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/product/parity_check_tensor.yml.