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AIMS Mathematics

2025, Volume 10, Issue 10: 24730-24754. doi: 10.3934/math.20251096
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Research article

Two classes of nearly optimal codebooks from generalized bent $ \mathbb{Z}_{4} $-valued quadratic forms

  • 1.
    Faculty of Mathematics and Statistics, Hubei Engineering University, Xiaogan, Hubei 432000, China
  • 2.
    School of Information Science and Engineering, Shandong Normal University, Jinan, Shandong 250358, China
  • Published: 29 October 2025

  • MSC : 11T23, 11T71, 94A60, 94B05

  • Codebooks with small maximum cross-correlations are desirable in many fields, such as compressed sensing, direct spread code-division multiple-access (CDMA) systems, and space-time codes. The objective of this paper is the construction of codebooks. Based on the theory of $ \mathbb{Z}_{4} $-valued quadratic forms, we propose two classes of generalized bent functions over $ \mathbb{Z}_{4} $, and construct new families of codebooks from these functions. The codebooks obtained in this paper are nearly optimal with respect to the Welch bound, and could have a very small alphabet size, which is of importance in practical applications. Moreover, some Boolean bent functions are also derived.

    Citation: Junchao Zhou, Tingting Pang. Two classes of nearly optimal codebooks from generalized bent $ \mathbb{Z}_{4} $-valued quadratic forms[J]. AIMS Mathematics, 2025, 10(10): 24730-24754. doi: 10.3934/math.20251096

    Related Papers:

  • Codebooks with small maximum cross-correlations are desirable in many fields, such as compressed sensing, direct spread code-division multiple-access (CDMA) systems, and space-time codes. The objective of this paper is the construction of codebooks. Based on the theory of $ \mathbb{Z}_{4} $-valued quadratic forms, we propose two classes of generalized bent functions over $ \mathbb{Z}_{4} $, and construct new families of codebooks from these functions. The codebooks obtained in this paper are nearly optimal with respect to the Welch bound, and could have a very small alphabet size, which is of importance in practical applications. Moreover, some Boolean bent functions are also derived.



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  • © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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