Prospects for Using Finite Algebraic Rings for Constructing Discrete Coordinate Systems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Theoretical basis of the paper seems solid, and I am particularly interested in its potential applications in the field of UAV swarm control and information security. The paper’s structure is fine, and it flows smoothly. However, it has several gaps, some unclear justifications, and more...

 

Paper jumps into mathematical formulations without properly explaining why this approach is necessary. You mention applications in UAV swarm control and information security, but these feel like an afterthought rather than an integral part of the research.

 

How does this method improve upon existing approaches? This part is missing.

 

You should improve the introduction section through discussing why this method is needed and how it compares to current solutions in UAV navigation/cryptography.

 

You argue that discrete coordinate systems can be useful for UAVs, but there’s no real demonstration of this. Have you tested this on an actual UAV system, or is this purely theoretical?

 

Paper moves from classical field extensions to algebraic rings but does not clearly justify why this shift is necessary. What I mean is: why use algebraic rings instead of Galois fields?

 

Derivations are fine however they lack intuitive explanations. I often find myself deciphering equations with little conceptual guidance. Please provide more intuition behind key formulas, suhc as, why idempotent elements are useful for defining a coordinate system?

 

Paper does not discuss how computationally expensive these algebraic operations are, especially when applied to large-scale systems (for instance UAV swarms).

 

Paper cites some relevant work but does not engage deeply with it. Paper could benefit from referencing the following study: “A Novel Swarm Unmanned Aerial Vehicle System: Incorporating Autonomous Flight, Real-Time Object Detection, and Coordinated Intelligence for Enhanced Performance.”

 

Paper cites some relevant work but does not engage deeply with it. To strengthen the discussion, please reference the following study: “A Novel Swarm Unmanned Aerial Vehicle System: Incorporating Autonomous Flight, Real-Time Object Detection, and Coordinated Intelligence for Enhanced Performance.”

 

The paper mentioned above has explored methods for optimizing real-time object detection, efficient task distribution, and decentralized networking to enhance UAV coordination. This aligns well with the motivation behind the proposed algebraic approach (both studies aim to improve UAV swarm intelligence and operational efficiency.)

 

The discussion mostly repeats the results instead of critically analyzing them. An important drawback!!

 

Consider addressing limitations. What are the risks or downsides of using this approach?

 

 

Comments on the Quality of English Language

The language is fine.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

In this paper, the method of non-standard algebraic extensions, which uses additional formal solutions of reduced equations, is expanded to three-dimensional space. This method generates algebraic rings instead of fields. It enables the creation of a discrete coordinate system where the basis vectors are represented by idempotent elements of the ring formed through this non-standard extension. The method discussed by the authors of this paper can provide new mathematical tools and methods for the application of UAV and other fields. Some comments are as follows.

[1] It is recommended to use standard mathematical methods to explain variables, for example, u₁ and u₂ in equation (1) are two variables, should not be explained by u_1,2, please check the full text.

[2] In the last paragraph of the introduction, the authors conclude: “In this paper, we propose a specific algorithm for constructing representations of the form (5), intended for displaying digital (discrete) coordinates, based on the non-standard method of algebraic extensions proposed in [45,46]. The difference is that in the cited works this method was applied to a two-dimensional space. In this paper, it is extended to the three-dimensional case.”

It is suggested that the authors add a discussion in the final paragraph of the introduction about the technical challenges of extending the method from two-dimensional to three-dimensional space, thereby introducing the main theoretical contributions of this paper.

[3] On page 6, the authors describe that the use of the minus sign in equation (16) is legitimate, among other things. What confuses the reviewer is that equation (16) represents the values of the sought parameters. What is the meaning of the minus sign in this context?

[4] It is recommended that the table be presented in the form of a three-line table.

[5] It is suggested that the authors discuss the challenges of the proposed method in UAV and other fields, such as the computational complexity of positioning and whether the accuracy of the discrete coordinate system can meet the demands of complex indoor and outdoor environments.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Good job on the revision.

Comments on the Quality of English Language

The language is fine.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have adequately addressed all my concerns, and I therefore recommend acceptance of this manuscript.