Error-Correcting Codes on Projective Bundles over Deligne–Lusztig Varieties

Round 1

Reviewer 1 Report

Dear Authors,

The article is applicable and well-written. There are some mistakes in English but it is acceptable at all.

The scientific soundness is good and I found it useful. I recommend this article. 

It can be improved!

Author Response

We thank the referee for the valuable comments. We have added some references in order that the reader has reachable information concerning the methods used to get our results. We have made wording revision as well and hope that the presentation is enhanced.

Reviewer 2 Report

Algebraic geometric codes have been an interesting area of research for many researchers. This beautiful paper studies error correcting codes on projective bundles over Deligne-Lusztig (DL) Varieties.  After collecting background on Deligne Lusztig varieties in Section 2, the construction of error correcting codes is given in Section 3. Parameters of codes coming from projective bundles over standard DL surfaces are obtained in next section. 

Although the results appears to be new, the writing of the paper requires improvements at many places so that the readership of the paper can be improved. Following corrections should be done before this paper can be accepted:

1. In Section 2, after giving definition authors should provide an example at page 2, specially lines 50 to 59 can be improved by providing citation such as for Lang-Steinberg Theorem (line 53) and lines 60-64. 

2. At many places whenever you introduced new term either give citation or give an example so that readership can be improved.

Author Response

We thank the referee for the valuable comments. We have added citation for the Lang-Steinberg Theorem and after Definition 1 we provide an example of a Deligne-Lusztig variety that we hope it illustrates this definition and help with the motivation of the introduced results. We have also added some references through the manuscript and we end the paper with specific examples of the applicability of these codes.

Reviewer 3 Report

Summary:

This paper provides lower bounds on the parameters of algebraic geometric error-correcting codes constructed from projective bundles over Deligne-Lusztig surfaces. The methods used in this article are based on intensive use of intersection theory, which allows for the extension of codes previously constructed from higher-dimensional varieties and curves. The paper also offers explicit examples of error-correcting codes constructed from projective bundles over Deligne-Lusztig surfaces. Overall, the paper contributes to the study of algebraic geometry codes defined over varieties in general, with a focus on Deligne-Lusztig varieties.

Strength:

- This paper mentions that the author S. Hansen applies general methods to obtain lower bounds, which could be considered innovative.

- This paper discusses the use of algebraic geometry codes to correct errors in data transmission and provides a theoretical analysis of the properties of these codes.

- This paper provides explicit examples of error-correcting codes constructed from surfaces of type A2 and A4, which could be valuable to readers interested in applying these codes in practice.

Weakness:

- No specific details are provided about the general methods to obtain lower bounds by S. Hansen.

- No specific empirical findings or results of the research. 

- This paper does not explicitly state the main question posed in the research or the specific conclusions drawn from the evidence and arguments presented.

- Though the paper describes the construction of algebraic geometry codes and provides examples, it does not discuss the practical application or potential real-world implications of these codes.

Details:

The main question addressed by the research is how to construct algebraic geometric error-correcting codes on projective bundles over Deligne-Lusztig surfaces and obtain lower bounds on their associated parameters.

The paper aims to extend the existing knowledge by providing lower bounds on the parameters of these codes and exploring their properties. It also mentions the potential for further research, such as extending the results to projective bundles of higher rank and gaining a more explicit understanding of the morphism involved.

This paper extends the methods and results presented in previous works such as J. Little and S. Hansen. It also provides explicit examples of error-correcting codes constructed from surfaces of type A2 and A4. However, not many details and empirical findings are provided in the paper.

readable but can be improved

Author Response

We thank the referee for the valuable comments. Regarding the general methods proposed by S. Hansen to obtain bounds for codes defined over higher dimensional varieties, we just mention those concerning the case when the points are included in a curve (Proposition 1 and Corollary 1) given that the other methods introduced in S. Hansen article are not used to prove our results.

We have added some examples at the end of the paper for the binary case that provide information on the applicability of these codes. 

Some specific conclusions have been added in order to highlight the introduced results as well as their applicability. 

We have made a wording revision and hope that the presentation has been enhanced.