Abstract: Codebooks are signal sets with low mutual correlation. Those codebooks meeting the Welch bound or the Lev-enshtein bound are mainly used to distinguish signals from different users in code-division multiple -access(CDMA) systems, and can also be applied in compressed sensing, coding theory, and quantum computing. Thispaper presents two new constructions of optimal and asymptotically optimal codebooks with respect to the Lev-enshtein bound. Firstly, a new class of optimal codebooks with respect to the Levenshtein bound is constructedusing design-theoretic object nets and Hadamard matrices. Secondly, a new class of asymptotically optimal codebooks with respect to the Levenshtein bound is constructed using permutation functions over finite fields.Parameter comparisons demonstrate that the construction parameters and methods for these two classes ofcodebooks represent novel contributions.

Key words: optimal codebook, asymptotically optimal codebook, Levenshtein bound, design-theoretic object net, permutation function, finite field