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Bivariate bicycle code

A translation-invariant CSS qLDPC family built from two sparse binary polynomials on a finite two-dimensional torus. Each polynomial becomes a cyclic-shift matrix; placing the two matrices in H_X=[A|B] and the swapped transposes in H_Z=[B^T|A^T] makes commutation follow from the abelian polynomial algebra. Representative finite blocks trade planar nearest-neighbor layout for a second coupling layer and encode many logical qubits at moderate distance.