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Gaussian sums, hyper Eisenstein sums and Jacobi sums over a local ring and their applications

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

It is well known that any finite commutative ring is isomorphic to a direct product of local rings via the Chinese remainder theorem. Hence, there is a great significance to the study of character sums over local rings. Character sums over finite rings have applications that are analogous to the applications of character sums over finite fields. In particular, character sums over local rings have many applications in algebraic coding theory. In this paper, we firstly present an explicit description on additive characters and multiplicative characters over a certain local ring. Then we study Gaussian sums, hyper Eisenstein sums and Jacobi sums over a certain local ring and explore their properties. It is worth mentioning that we are the first to define Eisenstein sums and Jacobi sums over this local ring. Moreover, we present a connection between hyper Eisenstein sums over this local ring and Gaussian sums over finite fields, which allows us to give the absolute value of hyper Eisenstein sums over this local ring. As an application, several classes of codebooks with new parameters are presented.

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Acknowledgements

The authors deeply thank Prof. Teo Mora and the anonymous reviewers for their valuable comments which have highly improved the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, Jiangsu, China

    Liqin Qian & Xiwang Cao

  2. Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100042, China

    Xiwang Cao

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  1. Liqin Qian
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  2. Xiwang Cao
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Correspondence to Liqin Qian.

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Qian, L., Cao, X. Gaussian sums, hyper Eisenstein sums and Jacobi sums over a local ring and their applications. AAECC 34, 211–244 (2023). https://doi.org/10.1007/s00200-021-00491-x

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