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We introduce a "hyperbicycle" ansatz for quantum codes which gives thehypergraph-product (generalized toric) codes by Tillich and Z\'emor andgeneralized bicycle codes by MacKay et al. as limiting cases. The constructionallows for both the lower and the upper bounds on the minimum distance; theyscale as a square root of the block length. Many of thus defined codes havefinite rate and a limited-weight stabilizer generators, an analog of classicallow-density parity check (LDPC) codes. Compared to the hypergraph-productcodes, hyperbicycle codes generally have wider range of parameters; inparticular, they can have higher rate while preserving the (estimated) errorthreshold.

Quantum "hyperbicycle" low-density parity check codes with finite rate