# New constructions of asymptotically optimal codebooks via character sums over a local ring

## TL;DR

This paper introduces new explicit constructions of complex codebooks over a local ring that are asymptotically optimal according to the Welch bound, utilizing character sums and providing novel parameters.

## Contribution

The paper provides the first explicit descriptions of characters and Gauss sums over a local ring and uses them to construct new asymptotically optimal codebooks.

## Key findings

- Two new families of codebooks achieve asymptotic optimality.
- Codebooks have novel parameters not previously reported.
- Explicit character sum descriptions facilitate codebook construction.

## Abstract

In this paper, we present explicit description on the additive characters, multiplicative characters and Gauss sums over a local ring. As an application, based on the additive characters and multiplicative characters satisfying certain conditions, two new constructions of complex codebooks over a local ring are introduced. With these two constructions, we obtain two families of codebooks achieving asymptotically optimal with respect to the Welch bound. It's worth mentioning that the codebooks constructed in this paper have new parameters.