Sheida Rabeti

Sheida Rabeti, Hessam Mahdavifar

We introduce univariate bicycle (UB) codes, a structured subclass of generalized bicycle (GB) quantum low-density parity-check (LDPC) codes obtained via a Frobenius relation. This construction reduces the code design space from a two-polynomial search in GB codes to a single-polynomial search, while preserving sparsity. We provide an explicit algebraic characterization of the logical coset spaces by constructing a basis for the logical quotient space, yielding a complete parametrization of logical operators. Leveraging this structure, we derive upper bounds on the minimum distance by relating structured logical representatives to cycle-density properties of associated circulant matrices. Finally, simulation results for short- to medium-length UB codes (block lengths ranging from a few hundred to approximately $10^3$) demonstrate competitive performance relative to existing GB and bivariate bicycle (BB) codes despite the additional algebraic restriction.

Sheida Rabeti, Hessam Mahdavifar

In this paper, we propose a belief-propagation (BP)-based decoder, termed the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for quantum low-density parity-check (QLDPC) codes. The key idea is to generate \emph{structured decoding diversity} by constructing multiple redundant parity-check representations via cycle-free subtree decompositions of the Tanner graph, and running BP decoding in parallel across these representations. This extends the classical Multiple-Bases Belief-Propagation (MBBP) framework to the quantum setting while preserving the linear-time complexity and efficiency of standard BP decoding, and avoids the need for super-linear post-processing. Simulation results demonstrate that MBBP-LD improves upon existing BP-based decoders, including BP with ordered statistics decoding (BP-OSD) and belief propagation with guided decimation (BPGD) across several QLDPC codes, while requiring substantially fewer total BP iterations. For bivariate bicycle codes $[[144,12,12]]$ and $[[288,12,18]]$, MBBP-LD achieves up to $20\%$ reduction in error rate compared to BPGD and up to $30\%$ compared to BP-OSD in the low- and moderate-error regimes. For the larger B1 code $[[882, 24, 18 \leq d \leq 24]]$, MBBP-LD attains comparable or improved performance relative to BPGD while maintaining BP-like decoding latency under parallel implementation.