In most practical quantum mechanical systems,quantum noise due to decoherence is highly biased towards dephas-ing.The quantum state suffers from phase flip noise much more seriously than from the bit flip noise.In this work,we construct new families of asymmetric quantum concatenated codes (AQCCs) to deal with such biased quantum noise.Our construction is based on a novel concatenation scheme for constructing AQCCs with large asymmetries,in which classical tensor product codes and concatenated codes are utilized to correct phase flip noise and bit flip noise,respectively.We generalize the original concatenation scheme to a more general case for better correcting degenerate errors.Moreover,we focus on constructing nonbinary AQCCs that are highly degenerate.Compared to previous literatures,AQCCs constructed in this paper show much better parameter performance than existed ones.Furthermore,we design the specific encoding circuit of the AQCCs.It is shown that our codes can be encoded more efficiently than standard quantum codes.