Skip to content
DraconiticResearch
Search Sources API
201,818 studies31 sourcesUpdated 51 min ago
PreprintarXiv licencearXiv

Rate-2/3 Girth-8 (3,18)-Regular Quantum LDPC Codes from Two-Branch Finite-Field Bases and CPM Lifts

  • Koki Okada
  • Kenta Kasai
Open at sourcePDFDOICite

Abstract

We construct a rate-2/3 quantum low-density parity-check (LDPC) code from a (3,18)-regular two-branch finite-field base and a circulant-permutation-matrix (CPM) lift of degree P=101. The resulting code is a Calderbank--Shor--Steane (CSS) code with parameters $[[34542,23032,18]]$. Its distance is established by a computer-assisted proof: an explicit nontrivial logical operator gives the upper bound, and a complete symmetry-reduced enumeration excludes every nonzero vector of weight below 18 in both parity-check kernels. We also prove that every affine-in-t CPM lift of the same base satisfying CSS orthogonality has distance at most 18 for every prime lift degree P>19. The construction has row weight 18 and column weight 3, and the Tanner graphs of HX and HZ separately have girth 8. Decoder experiments with log-likelihood-ratio (LLR) joint belief propagation (BP) and deterministic post-processing show no failures in 108 trials at p=0.01, and a finite-length frame error rate (FER) sweep estimates the transition near p=0.029.

Related

  1. Whither quantum advantage? New classical lower bounds for Max-Cut on large-girth 3-regular graphs
    Flaviano Morone, Shivang Rawat, Dries Sels
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsCondensed matterarXiv
  2. A Two-Branch Finite-Field Construction for Regular CSS LDPC Bases
    Koki Okada, Kenta Kasai
    arXivPreprintCC BY 4.0Full textGeneral mathematicsQuantum physics & quantum informationarXiv
  3. Low-Density Parity-Check Codes of High Girth from Permutation Polynomials
    Tasawwar Hussain, Anthony Gómez-Fonseca, Tefjol Pllaha
    arXivPreprintCC BY 4.0Full textGeneral mathematicsarXiv
  4. High-Girth QLDPC Codes from Circulant and Affine Permutation Matrices
    Vahid Nourozi, Roxana Smarandache, David G. M. Mitchell†
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Quantum MDS codes from complements of unions of finite-field subsets
    Naihong Hu, Hong Ji
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryMathematical physicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{okada2026rate23,
  title = {{Rate-2/3 Girth-8 (3,18)-Regular Quantum LDPC Codes from Two-Branch Finite-Field Bases and CPM Lifts}},
  author = {Koki Okada and Kenta Kasai},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.27130},
  eprint = {2606.27130},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.27130},
}
RIS
TY  - UNPB
TI  - Rate-2/3 Girth-8 (3,18)-Regular Quantum LDPC Codes from Two-Branch Finite-Field Bases and CPM Lifts
AU  - Koki Okada
AU  - Kenta Kasai
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2606.27130
UR  - https://arxiv.org/abs/2606.27130
AB  - We construct a rate-2/3 quantum low-density parity-check (LDPC) code from a (3,18)-regular two-branch finite-field base and a circulant-permutation-matrix (CPM) lift of degree P=101. The resulting code is a Calderbank--Shor--Steane (CSS) code with parameters $[[34542,23032,18]]$. Its distance is established by a computer-assisted proof: an explicit nontrivial logical operator gives the upper bound, and a complete symmetry-reduced enumeration excludes every nonzero vector of weight below 18 in both parity-check kernels. We also prove that every affine-in-t CPM lift of the same base satisfying CSS orthogonality has distance at most 18 for every prime lift degree P>19. The construction has row weight 18 and column weight 3, and the Tanner graphs of H_X and H_Z separately have girth 8. Decoder experiments with log-likelihood-ratio (LLR) joint belief propagation (BP) and deterministic post-processing show no failures in 10^8 trials at p=0.01, and a finite-length frame error rate (FER) sweep estimates the transition near p=0.029.
ER  -