We present a modified version of the Bravyi-Terhal bound that applies to quantum codes defined by local parity-check constraints on aD-dimensional lattice quotient. Specifically, we consider a quotient ZD/Λ of ZDof cardinality ℓ, where Λ is someD-dimensional sublattice of ZD: we suppose that every vertex of this quotient indexesmqubits of a stabilizer codeC, which therefore has lengthn=mℓ. We prove that if all stabilizer generators act on qubits whose indices lie within a ball of radius ρ, then the minimum distancedof the code satisfiesd≤m√ γD( √D+ 4ρ)ℓD−1/D, where γDis theD-dimensional Hermite constant. We then apply this bound to derive an upper bound on the minimum distance of Abelian Two-Block Group Algebra (2BGA) codes whose parity-check matrices have the form [A|B] with each submatrix representing an element of a group algebra over a finite abelian group.

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 [[2n2, 2,n]], [[4r2, 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.