  We study the performance of medium-length quantum LDPC (QLDPC) codes in the
depolarizing channel. Only degenerate codes with the maximal stabilizer weight
much smaller than their minimum distance are considered. It is shown that with
the help of OSD-like post-processing the performance of the standard belief
propagation (BP) decoder on many QLDPC codes can be improved by several orders
of magnitude. Using this new BP-OSD decoder we study the performance of several
known classes of degenerate QLDPC codes including hypergraph product codes,
hyperbicycle codes, homological product codes, and Haah's cubic codes. We also
construct several interesting examples of short generalized bicycle codes. Some
of them have an additional property that their syndromes are protected by small
BCH codes, which may be useful for the fault-tolerant syndrome measurement. We
also propose a new large family of QLDPC codes that contains the class of
hypergraph product codes, where one of the used parity-check matrices is
square. It is shown that in some cases such codes have better performance than
hypergraph product codes. Finally, we demonstrate that the performance of the
proposed BP-OSD decoder for some of the constructed codes is better than for a
relatively large surface code decoded by a near-optimal decoder.
