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In this paper, we utilize a concatenation scheme to construct new families of quantum error correction codes achieving the quantum Gilbert-Varshamov (GV) bound asymptotically. Weconcatenate alternant codes with any linear code achievingthe classical GV bound to construct Calderbank-Shor-Steane (CSS) codes. We show that the concatenated code can achieve the quantum GV bound asymptotically and can approach the Hashing bound for asymmetric Pauli channels. By combing Steane&#x2019;s enlargement construction of CSS codes, we derive a family of enlarged stabilizer codes achieving the quantum GV bound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parameters    $\mathscr {Q}_{1}=[[N,\Omega (\sqrt {N}),\Omega (\sqrt {N})]]$   , and    $\mathscr {Q}_{2}=[[N,\Omega (N/\log N),\Omega (N/\log N)/\Omega (\log N)]]$   . We show that    $\mathscr {Q}_{1}$    can be encoded very efficiently by circuits of size    $O(N)$    and depth    $O(\sqrt {N})$   . For an input error syndrome,    $\mathscr {Q}_{1}$    can correct any adversarial error of weight up to half the minimum distance bound in    $O(N)$    time.    $\mathscr {Q}_{1}$    can also be decoded in parallel in    $O(\sqrt {N})$    time by using    $O(\sqrt {N})$    classical processors. For an input error syndrome, we proved that    $\mathscr {Q}_{2}$    can correct a linear number of    ${X}$   -errors with high probability and an almost linear number of    ${Z}$   -errors in    $O(N)$    time. Moreover,    $\mathscr {Q}_{2}$    can be decoded in parallel in    $O(\log (N))$    time by using    $O(N)$    classical processors.

Partially Concatenated Calderbank-Shor-Steane Codes Achieving the Quantum Gilbert-Varshamov Bound Asymptotically