On Intersection Codes and Iterative Decoding

Martin Bossert, F. Hergert

Tzu‐Hao Lin, Ching–Yi Lai

In this paper, we introduce the Union-Intersection Union-Find (UIUF) algorithm for decoding depolarizing errors in topological codes, combining the strengths of iterative and standard Union-Find (UF) decoding. While iterative UF improves performance at moderate error rates, it lacks an error correction guarantee. To address this, we develop UIUF, which maintains the enhanced performance of iterative UF while ensuring error correction up to half the code distance. Through simulations under code capacity, phenomenological, and biased noise models, we show that UIUF significantly outperforms UF, reducing the logical error rate by over an order of magnitude (at around 10-5). Moreover, UIUF achieves lower logical error rates than the Minimum Weight Perfect Matching (MWPM) decoder on rotated surface codes under both the code capacity and phenomenological noise models, while preserving efficient linear-time complexity.

P. Heijnen, F.M.J. Willems, T.J. Tjalkens

Gretchen L. Matthews, Julia Shapiro

In 2025, Jordan et al. introduced Decoded Quantum Interferometry (DQI), a quantum algorithm for combinatorial optimization based on decoding. They considered Reed-Solomon decoding and its associated optimization problem, called Optimal Polynomial Intersection (OPI), which may be viewed as a polynomial regression problem over a finite field. DQI exhibits provable speedups on certain problem instances and establishes a connection between decoding problems and optimization tasks. Leveraging the well-understood dual structure and decoding theory of algebraic geometry codes from other curve families, we introduce COFI: Curve-based Optimal Function Intersection. By considering two-point Hermitian codes, Suzuki codes, and extended norm-trace codes, we broaden the range of algebraic geometry codes used in DQI and identify families that offer further improvements over one-point Hermitian codes in the Hermitian Optimal Polynomial Intersection framework considered by Jordan and Gu. Depending on the family and parameter regime, these curves can reduce quantum resource requirements or increase the number of constraints that can be considered.

O. Pothier, Loïc Brunel, Joseph J. Boutros

We describe a class of asymptotically good codes built from the intersection of randomly permuted binary BCH codes. This family of pseudo-random error correcting codes, called generalized low density (GLD) codes, is a direct generalization of Gallager's low density parity check (LDPC) codes. GLD codes belong to the larger family of Tanner codes based on a random bipartite graph. We study the GLD ensemble performance and prove the asymptotically good property. We also compare GLD codes minimum distance and performance to the Varshamov-Gilbert bound and BSC capacity respectively. The results show that maximum-likelihood decoding of GLD codes achieves near capacity efficiency. The suboptimal iterative decoding of GLD codes is briefly presented. Experimental results of small and large blocklength codes are finally illustrated on both AWGN and Rayleigh fading channels.

Pascal O. Vontobel