Optimal codebooks achieving the Levenshtein bound from generalized bent functions over ℤ4$\mathbb {Z}_{4}$

Codebooks (also called signal sets) meeting the Welch bound or the Levenshtein bound are used to distinguish among the signals of different users in CDMA systems. Recently, Zhou, Ding and Li proposed a construction of optimal codebooks based on a set of bent functions satisfying certain conditions over finite fields. In this paper, inspired by the work of Zhou, Ding and Li, we introduce a new construction of codebooks from generalized Boolean bent functions over ℤ4$\mathbb {Z}_{4}$. Using this construction, we obtain some optimal codebooks achieving the Levenshtein bound. The codebooks constructed in this paper have parameters (22m+2m, 2m) and a small alphabet size 6. In particular, a set of generalized Boolean bent functions satisfying certain conditions is constructed. As byproducts, some Boolean bent functions are derived.

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