  In this paper, we present a new construction of asymmetric quantum codes
(AQCs) by combining classical concatenated codes (CCs) with tensor product
codes (TPCs), called asymmetric quantum concatenated and tensor product codes
(AQCTPCs) which have the following three advantages. First, only the outer
codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes,
and any classical linear code can be used for the inner, which makes AQCTPCs
very easy to construct. Second, most AQCTPCs are highly degenerate, which means
they can correct many more errors than their classical TPC counterparts.
Consequently, we construct several families of AQCs with better parameters than
known results in the literature. Third, AQCTPCs can be efficiently decoded
although they are degenerate, provided that the inner and outer codes are
efficiently decodable. In particular, we significantly reduce the inner
decoding complexity of TPCs from $\Omega(n_2a^{n_1})(a>1)$ to $O(n_2)$ by
considering error degeneracy, where $n_1$ and $n_2$ are the block length of the
inner code and the outer code, respectively. Furthermore, we generalize our
concatenation scheme by using the generalized CCs and TPCs correspondingly.
