  Codebooks with small inner-product correlation have application in unitary
space-time modulations, multiple description coding over erasure channels,
direct spread code division multiple access communications, compressed sensing,
and coding theory. It is interesting to construct codebooks (asymptotically)
achieving the Welch bound or the Levenshtein bound. This paper presented a
class of generalized bent $\mathbb{Z}_4$-valued quadratic forms, which contain
functions of Heng and Yue (Optimal codebooks achieving the Levenshtein bound
from generalized bent functions over $\mathbb{Z}_4$. Cryptogr. Commun. 9(1),
41-53, 2017). By using these generalized bent $\mathbb{Z}_4$-valued quadratic
forms, we constructs optimal codebooks achieving the Levenshtein bound. These
codebooks have parameters $(2^{2m}+2^m,2^m)$ and alphabet size $6$.
