# Codebooks from generalized bent $\mathbb{Z}_4$-valued quadratic forms

## TL;DR

This paper introduces a new class of generalized bent $\

## Contribution

It constructs optimal codebooks achieving the Levenshtein bound using generalized bent $\

## Key findings

- Constructed codebooks with parameters $(2^{2m}+2^m,2^m)$ and alphabet size 6.
- Achieved codebooks meet the Levenshtein bound.
- Extended previous work on bent functions over $\

## Abstract

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 $6$.