# Six constructions of asymptotically optimal codebooks via the character sums

@article{Lu2019SixCO, title={Six constructions of asymptotically optimal codebooks via the character sums}, author={Wenlong Lu and Xia Wu and Xiwang Cao and Ming Chen}, journal={Designs, Codes and Cryptography}, year={2019}, volume={88}, pages={1139-1158} }

In this paper, using additive characters of finite field, we find a codebook which is equivalent to the measurement matrix in Mohades et al. (IEEE Signal Process Lett 21(7):839–843, 2014). The advantage of our construction is that it can be generalized naturally to construct the other five classes of codebooks using additive and multiplicative characters of finite field. We determine the maximum cross-correlation amplitude of these codebooks by the properties of characters and character sums…

## 11 Citations

### Four Constructions of Asymptotically Optimal Codebooks via Additive Characters and Multiplicative Characters

- Mathematics
- 2019

: In this paper, we present four new constructions of complex codebooks with multiplicative characters, additive characters, and quadratic irreducible polynomials and determine the maximal…

### Four Constructions of Asymptotically Optimal Codebooks via Additive Characters and Multiplicative Characters

- Mathematics, Computer Science
- 2019

Four new constructions of complex codebooks with multiplicative characters, additive characters, and quadratic irreducible polynomials are presented and it is proved that the codebooks constructed are asymptotically optimal with respect to the Welch bound.

### Two constructions of asymptotically optimal codebooks via the trace functions

- Computer Science, MathematicsCryptography and Communications
- 2020

It is proved that the codebooks constructed are asymptotically optimal with respect to the Welch bound, and the maximal cross-correlation amplitude of these codebooks is determined.

### Constructions of asymptotically optimal codebooks with respect to Welch bound and Levenshtein bound

- Computer ScienceAdvances in Mathematics of Communications
- 2021

Several classes of codebooks are introduced, whose maximum cross-correlation amplitudes asymptotically achieve the corresponding Welch bound and Levenshtein bound.

### The Jacobi sums over Galois rings of arbitrary characters and their applications in constructing asymptotically optimal codebooks

- MathematicsArXiv
- 2021

The deterministic construction of codebooks based on the Jacobi sums over Galois rings of arbitrary characteristics is presented, which produces asymptotically optimal codebooks with respect to the Welch bound.

### Nearly optimal codebooks from generalized Boolean bent functions over ℤ4

- Computer ScienceAdv. Math. Commun.
- 2022

The codebooks constructed in this paper are nearly optimal with respect to the Welch bound, and their parameters are new.

### Certain codebooks and the generalized Erdős-Falconer distance problem ( extended abstract )

- Mathematics
- 2020

The main purpose of this extended abstract is to establish a connection between minimizing the maximal cross-correlation amplitude of certain codebooks and the generalized Erdős-Falconer distance…

### Three constructions of asymptotically optimal codebooks via multiplicative characters of finite fields

- MathematicsAdvances in Mathematics of Communications
- 2022

### Character sums over a non-chain ring and their applications

- Mathematics, Computer ScienceAdvances in Mathematics of Communications
- 2021

This paper studies character sums over a certain non-chain ring and their applications in codebooks and presents three classes of asymptotically optimal code books with respect to the Welch bound and a family of optimal codebooks withrespect to the Levenshtein bound.

## References

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It is proved that the codebooks constructed are asymptotically optimal with respect to the Welch bound, and the maximal cross-correlation amplitude of these codebooks is determined.

### Constructions of Codebooks Asymptotically Achieving the Welch Bound With Additive Characters

- Computer ScienceIEEE Signal Processing Letters
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Following the proposed constructions, two classes of asymptotically optimal codebooks with respect to the Welch bound are obtained and the resulting codebooks could have a small alphabet size.

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A new character sum, the hyper Eisenstein sum, is presented and the properties of this character sum are studied and the codebooks generated by these constructions asymptotically meet the Welch bound.

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In this paper, two new constructions of infinitely many codebooks with multiplicative characters of finite fields are presented, which produce complex codebooks asymptotically achieving the Levenshtein bound and code books asymptic achieving the Welch bound.

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This paper proposes new constructions of codebooks nearly meeting the Welch bound with equality using difference sets and the product of Abelian groups, which yield many codebooks with new parameters.

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The complex codebooks produced by these two constructions are proved to be asymptotically optimal with respect to the Welch bound.

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This work constructs analytically optimal codebooks meeting the Welch lower bound, and develops an efficient numerical search method based on a generalized Lloyd algorithm that leads to considerable improvement on the achieved I/sub max/ over existing alternatives.

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It is proved that the maximum magnitude of inner products between the code vectors of the proposed codebooks asymptotically achieves the Welch bound equality for sufficiently large p and derive their inner product distribution.

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The objectives of this correspondence are to present a generic construction of MWBE codebooks that contains the previous two constructions as special cases and describe new MWBEcodebooks that cannot be produced by the earlier two construction.