(2,2)-GB Codes: Classification and Comparison With Weight-4 Surface Codes

Abstract

Generalized Bicycle (GB) codes offer a compelling alternative to surface codes for quantum error correction. This paper focuses on (2,2)-Generalized Bicycle codes, constructed from pairs of binary circulant matrices with two non-zero elements per row. Leveraging a lower bound on their minimum distance, we construct three novel infinite families of optimal (2,2)-GB codes with parameters $ [[2n^{2}, 2, n]] $ , $ [[4r^{2}, 2, 2r]] $ , and $ [[(2t + 1)^{2} + 1, 2, 2t + 1]] $ . These families match the performance of Kitaev’s toric code and the best 2D weight-4 surface codes, reaching known theoretical limits. In particular, the second family breaks a long-held belief by providing optimal even-distance GB codes, previously deemed impossible. All are CSS codes derived from Cayley graphs. Recognizing that standard equivalence relations do not preserve their CSS structure, we introduce a CSS-preserving equivalence relation for rigorous comparison of Cayley graph-based CSS codes. Under this framework, the first two families are inequivalent to all previously known optimal weight-4 2D surface codes, while the third family is equivalent to the best-known odd-distance 2D surface code. Finally, we classify all extremal, non-equivalent $(2,2)$ -GB codes with length below 200 and present a comparison table with existing notable 2D weight-4 surface codes.