1. Character sums over a non-chain ring and their applications
- Author
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Liqin Qian and Xiwang Cao
- Subjects
Ring (mathematics) ,Algebra and Number Theory ,Computer Networks and Communications ,Applied Mathematics ,Gaussian ,Codebook ,Coding theory ,Microbiology ,Combinatorics ,Character sum ,symbols.namesake ,Asymptotically optimal algorithm ,Character (mathematics) ,Chain (algebraic topology) ,symbols ,Discrete Mathematics and Combinatorics ,Mathematics - Abstract
Some valuable results over rings have a promising utilization in coding theory and error-correcting code theory. In this paper, we study character sums over a certain non-chain ring and their applications in codebooks. There are two major ingredients in this study. The first ingredient is to investigate Gaussian sums, hyper Eisenstein sums, Jacobi sums over a certain non-chain ring and study the properties of these character sums. For their applications, the second ingredient is to present three classes of asymptotically optimal codebooks with respect to the Welch bound and a family of optimal codebooks with respect to the Levenshtein bound, which are constructed from character sums over a certain non-chain ring.
- Published
- 2023