Ancestor Lifting of Quasi-Cyclic Additive Codes and Certified Quantum Codes
Abstract
Additive codes over Ffour that are self-orthogonal under the trace inner product correspond to binary quantum stabilizer codes. We study 1-generator (1,2)-quasi-cyclic (QC) additive codes and their exact quantum distances. For composite lengths n=ℓ m, an ancestor-lifting theorem transports dual words of the folded ancestor to the length-n code. A two-layer decomposition for ℓ=2 and a CRT splitting for odd ℓ yield dimension laws and explain equality, escape, and mixed-branch behavior in the certified examples. A palindromic criterion ensures symplectic self-orthogonality, while reversal pairing and a symplectic Gram rank formula describe broader construction families. Exact distances are established by information-set decoding screening, Brouwer--Zimmermann computation on the additive dual, and independent witness verification. We certify twenty-one record-breaking binary quantum codes across twenty entries of Grassl's code table, with twenty-eight certified members in total. Propagation improves thirty-four further entries, and explicit constructions are supplied for three table-tying GV entries marked Code Missing Construction. Six additional candidates are reported only with rigorous upper bounds and heuristic screening evidence. The results connect divisor-lattice structure in additive quasi-cyclic codes with reproducible certification of exact quantum distances.
Related
- Weak and Serre Lifting of Cyclic Modules and the Liftability of Auslander Transposes
- Carlet's cyclic-additive conjecture for the Kasami monomials
- Lifting an [n,k,d] Binary Linear Code to an $[[n,k,d]]$ Quantum Stabilizer Code
- Certified Non-Degenerate Maximum-Likelihood Benchmarking of Quantum LDPC Decoders
- Certified bounds on optimization problems in quantum theory
BibTeX
@misc{lu2026ancestor,
title = {{Ancestor Lifting of Quasi-Cyclic Additive Codes and Certified Quantum Codes}},
author = {Liangdong Lu and Ran Xu and Ruipan Yang and Qiang Fu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36700},
eprint = {2610.36700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36700},
}
RIS
TY - UNPB
TI - Ancestor Lifting of Quasi-Cyclic Additive Codes and Certified Quantum Codes
AU - Liangdong Lu
AU - Ran Xu
AU - Ruipan Yang
AU - Qiang Fu
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36700
UR - https://arxiv.org/abs/2610.36700
AB - Additive codes over Ffour that are self-orthogonal under the trace inner product correspond to binary quantum stabilizer codes. We study 1-generator (1,2)-quasi-cyclic (QC) additive codes and their exact quantum distances. For composite lengths n=ℓ m, an ancestor-lifting theorem transports dual words of the folded ancestor to the length-n code. A two-layer decomposition for ℓ=2 and a CRT splitting for odd ℓ yield dimension laws and explain equality, escape, and mixed-branch behavior in the certified examples. A palindromic criterion ensures symplectic self-orthogonality, while reversal pairing and a symplectic Gram rank formula describe broader construction families. Exact distances are established by information-set decoding screening, Brouwer--Zimmermann computation on the additive dual, and independent witness verification. We certify twenty-one record-breaking binary quantum codes across twenty entries of Grassl's code table, with twenty-eight certified members in total. Propagation improves thirty-four further entries, and explicit constructions are supplied for three table-tying GV entries marked Code Missing Construction. Six additional candidates are reported only with rigorous upper bounds and heuristic screening evidence. The results connect divisor-lattice structure in additive quasi-cyclic codes with reproducible certification of exact quantum distances.
ER -