High-Girth QLDPC Codes from Circulant and Affine Permutation Matrices

Abstract

We study two explicit constructions of binary Calderbank--Shor--Steane quantum low-density parity-check (QLDPC) codes with check weight six. The first uses a fixed circulant shift pattern and yields parameters code6N2N+26 with Tanner graph girth twelve for N=31 and every N≥37. A graph incidence interpretation gives the ranks and girths, and a uniform weight six logical operator establishes the exact distance. The second construction uses fixed matrix polynomials with circulant and affine permutation coefficients. Here, a polynomial commutation identity guarantees orthogonality for every cyclic lift. Four realizations give $[[192,16,8]]$, $[[288,16,12]]$, $[[384,16,14]]$, and $[[576,32,20]]$, with Tanner girth eight and computationally certified distances. We give reproducible construction data, distinguish classical cycle distance from quantum distance, and identify the limitations of both families.

Low-Density Parity-Check Codes of High Girth from Permutation Polynomials
Tasawwar Hussain, Anthony Gómez-Fonseca, Tefjol Pllaha
arXiv2 Oct 2026PreprintCC BY 4.0Full textGeneral mathematicsarXiv
Understanding Circulant Permutation to Extend the CMinHash Estimator
Keegan Kang, Avery Hood, Benedict H. W. Wong
arXiv13 Oct 2026PreprintarXiv licenceFull textStatisticsarXiv
On properties of the conformal circulant matrices
Cristina Manzaneda, Enide Andrade
arXiv13 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEs Combinatorics & discrete mathematicsarXiv
Freeness for the G-circulant Decomposition of the Partial Transpose of Random Matrices
Octavio Arizmendi, Julian Zazueta-Obeso
arXiv9 Sep 2026PreprintCC BY 4.0Full textAnalysis & PDEs ProbabilityarXiv
Revisiting Toeplitz and Hankel random matrices via *-convergence of circulant-type matrices
Arup Bose, Pradeep Vishwakarma
arXiv17 Sep 2026PreprintarXiv licenceFull textProbabilityarXiv

Cite

BibTeX RIS

BibTeX

@misc{nourozi2026highgirth,
  title = {{High-Girth QLDPC Codes from Circulant and Affine Permutation Matrices}},
  author = {Vahid Nourozi and Roxana Smarandache and David G. M. Mitchell†},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19451},
  eprint = {2610.19451},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19451},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - High-Girth QLDPC Codes from Circulant and Affine Permutation Matrices
AU  - Vahid Nourozi
AU  - Roxana Smarandache
AU  - David G. M. Mitchell†
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19451
UR  - https://arxiv.org/abs/2610.19451
AB  - We study two explicit constructions of binary Calderbank--Shor--Steane quantum low-density parity-check (QLDPC) codes with check weight six. The first uses a fixed circulant shift pattern and yields parameters code6N2N+26 with Tanner graph girth twelve for N=31 and every N≥37. A graph incidence interpretation gives the ranks and girths, and a uniform weight six logical operator establishes the exact distance. The second construction uses fixed matrix polynomials with circulant and affine permutation coefficients. Here, a polynomial commutation identity guarantees orthogonality for every cyclic lift. Four realizations give $[[192,16,8]]$, $[[288,16,12]]$, $[[384,16,14]]$, and $[[576,32,20]]$, with Tanner girth eight and computationally certified distances. We give reproducible construction data, distinguish classical cycle distance from quantum distance, and identify the limitations of both families.
ER  -

Abstract

Related

Cite