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Published in: Applicable Algebra in Engineering, Communication and Computing 2/2023

05-03-2021 | Original Paper

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

Authors: Liqin Qian, Xiwang Cao

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 2/2023

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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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Metadata
Title
Gaussian sums, hyper Eisenstein sums and Jacobi sums over a local ring and their applications
Authors
Liqin Qian
Xiwang Cao
Publication date
05-03-2021
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 2/2023
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-021-00491-x

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