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We construct a new explicit family of good quantum low-density parity-checkcodes which additionally have linear time decoders. Our codes are based on athree-term chain $(\mathbb{F}_2^{m\times m})^V \quad\xrightarrow{\delta^0}\quad (\mathbb{F}_2^{m})^{E} \quad\xrightarrow{\delta^1}\quad \mathbb{F}_2^F$ where $V$ ($X$-checks) are the vertices, $E$ (qubits) arethe edges, and $F$ ($Z$-checks) are the squares of a left-right Cayley complex,and where the maps are defined based on a pair of constant-size random codes$C_A,C_B:\mathbb{F}_2^m\to\mathbb{F}_2^\Delta$ where $\Delta$ is the regularityof the underlying Cayley graphs.  One of the main ingredients in the analysis is a proof of anessentially-optimal robustness property for the tensor product of two randomcodes.

Good Quantum LDPC Codes with Linear Time Decoders