Quantum Bicycle LDPC Codes with High kd2/n from Divisor-Driven Search
Abstract
Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring F2[x]/(xl−1): self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over F4, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes [[42,12,4]]2 and [[62,12,4]]2 and produces a family of codes with competitive figure of merit kd2/n, including [[66,20,7]]2 with kd2/n=14.85, above the bivariate bicycle code [[144,12,12]]2 (kd2/n=12) at less than half the block length, together with [[46,2,8]]2, [[66,2,9]]2, [[66,4,8]]2, [[66,6,8]]2 and, at n=90, [[90,16,6]]2, [[90,18,6]]2, [[90,20,6]]2. An exhaustive census at n=48 delineates the boundary of this picture: we exhibit a [[48,10,6]]2 code from a minimal $48$-element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes [[48,10,5]]2 from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.
Related
- Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark
- Spectral Theory of Semisimple Bivariate Bicycle Codes
- Optimizing Encoder Circuits of Entanglement-Assisted Quantum LDPC Codes via Beam Search
- Existence and Characterization of Bivariate Bicycle Codes
- New lower bounds for binary constant-weight codes from prescribed-automorphism search, and two quantum codes meeting their upper bounds
BibTeX
@misc{lu2026quantum,
title = {{Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search}},
author = {Liangdong Lu and Guanmin Guo and Yang Liu and Ruipan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.09115},
eprint = {2608.09115},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09115},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Bicycle LDPC Codes with High kd^2/n from Divisor-Driven Search
AU - Liangdong Lu
AU - Guanmin Guo
AU - Yang Liu
AU - Ruipan Yang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.09115
UR - https://arxiv.org/abs/2608.09115
AB - Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring F_2[x]/(x^l−1): self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over F_4, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes [[42,12,4]]_2 and [[62,12,4]]_2 and produces a family of codes with competitive figure of merit kd^2/n, including [[66,20,7]]_2 with kd^2/n=14.85, above the bivariate bicycle code [[144,12,12]]_2 (kd^2/n=12) at less than half the block length, together with [[46,2,8]]_2, [[66,2,9]]_2, [[66,4,8]]_2, [[66,6,8]]_2 and, at n=90, [[90,16,6]]_2, [[90,18,6]]_2, [[90,20,6]]_2. An exhaustive census at n=48 delineates the boundary of this picture: we exhibit a [[48,10,6]]_2 code from a minimal $48$-element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes [[48,10,5]]_2 from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.
ER -