In this work, we propose a novel design of high-rate Calderbank--Shor--Steane (CSS) quantum low-density parity-check (QLDPC) codes based on quasi-dyadic matrices. The Tanner graphs associated with the two component classical codes used to construct the proposed CSS codes are characterized by a girth of at least $6$. We also derive an exact count of length-$4$ cycles in the quaternary representation of the parity-check matrix. Monte Carlo simulations under code-capacity and phenomenological noise models show that the proposed codes have competitive finite-length logical error rate performance compared with state-of-the-art codes in both these settings.

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