Abstract
Surface codes have historically been the dominant choice for quantum error correction due to their superior error threshold performance. However, recently, a new class of Generalized Bicycle (GB) codes, constructed from binary circulant matrices with three non-zero elements per row, achieved comparable performance with fewer physical qubits and higher encoding efficiency. In this article, we focus on a subclass of GB codes, which are constructed from pairs of binary circulant matrices with two non-zero elements per row. We introduce a family of codes that generalizes both standard and optimized Kitaev codes for which we have a lower bound on their minimum distance, ensuring performance better than standard Kitaev codes. These codes exhibit parameters of the form [| 2n , 2, ≥ √n |] where n is a factor of 1 + d² . For code lengths below 200, our analysis yields 21 codes, including 7 codes from Pryadko and Wang's database, and unveils 14 new codes with enhanced minimum distance compared to standard Kitaev codes. Among these, 3 surpass all previously known weight-4 GB codes for distances 4, 8, and 12.