Optimal Quaternary Hermitian LCD Codes

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Title: Optimal Quaternary Hermitian LCD codes

In this manuscript, the authors study quaternary Hermitian LCD codes. The research approach is trivial in my opinion. From already obtained codes located in databases and easily accessible with puncturing and shortening, the authors construct other LCD codes. For the calculations, they use the MAGMA software package and not their own methods and algorithms.

 The obtained results are also not of particular interest because in most cases there are already known codes with such parameters. There is no theoretical result either.

For these reasons, I cannot recommend this article for publication in the MDPI journal Entropy.

Comments on the Quality of English Language

No comments

Author Response

Dear Prof.  We would like to thank you for your valuable comments that have been helpful to improve the paper considerably. We hope to study new  quaternary Hermitian LCD codes  using new tools in the future. I look forward to having good academic interaction and exchange with you in the future. 

Reviewer 2 Report

Comments and Suggestions for Authors

Nowadays Hermitian LCD codes are rather important since they found applications in

cryptology and quantum error correcting coding.

In this paper, the authors systemized known constructions and presented several newly found codes.

Search results are summarized in a complete table of codes of length up to 25.

 

Remarks:

1. In Table 3, I recommend marking those codes that achieve bounds given in the Grassl table.

2. Typo: Table 3 has the wrong number (Table 2).

 

I recommend acceptance after minor corrections.

 

 

Author Response

We would like to thank you for your kind advises. 

We mark those codes that achieve bounds given in the Grassl table,

and revise the number of Table 3.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

I did not go through the proofs. However, it seems that all the results are new and interesting.

Comments on the Quality of English Language

I suggest the authors to check the language expression one more time.

Author Response

 We would like to thank you for    your valuable suggestions and  check the language expression carefully.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

In this manuscript, the authors  aim to find good quaternary Hermitian LCD codes.

   There are two things to do: to give quaternary Hermitian LCD codes 

   and verify they are optimal.  The strategy is clear. 

   The authors present the generator matrices of some LCD codes . 

   Then, the idea is  in the construction of  puncturing, shortening and

   combination  the given  codes to obtain more good quaternary Hermitian LCD 

   codes with different lengths. Results including  the best-known quaternary

   Hermitian LCD codes of any length $ n \leq 25$ with corresponding

   dimension $k$ are presented.

   

  

   {\bf Conclusions about the paper}

 

     The results are interesting and  correct. 

     In my opinion, this paper 

     can be accepted after considering some minor revisions. 

     Some specific comments are given 

     in the following lines. 

   

 

  

 

1. Page 1, line -10 : " Carlet and Guilley in$^{3}$

showed that LCD codes play an important role" $\rightarrow$ 

"Carlet and Guilley in Ref.$[3]$

showed that LCD codes play an important role". 

Modify  in other similar situations

 

2. Page 2, line -2 : "If there  exists  an $[n,k,d]$ and  $d$ is the largest value present known, then

$\mathcal{C}$ is called a {\it best known }  code and also 

denoted $[n,k,d_{o}(n,k)]$." $\rightarrow$ "For a given code $[n,k,d]$, 

if $d$ is the largest value present known, then

$\mathcal{C}$ is called a {\it best-known }  code and also 

denoted $[n,k,d_{o}(n,k)]$.".

 

3. Page 2, line -1 : "If an $\mathcal{C}=[n, k, d_{l}(n,k)]$ LCD

code is..." $\rightarrow$ 

"If a code  $\mathcal{C}=[n, k, d_{l}(n,k)]$ LCD

code is...". 

 

4. Page 3, line 5:  "$({ u,v})={ uv^{h}}=u_{1}\bar{v_{1}}+u_{2}\bar

{v_{2}}+\cdot\cdot\cdot+u_{n}\bar{v_{n}}.$" and "$({ u,v})_{h}={ uv^{2}}=u_{1}\bar{v_{1}}+u_{2}\bar

{v_{2}}+\cdot\cdot\cdot+u_{n}\bar{v_{n}}.$". 

 

5. Page 3, line 8: "If

$C$ is a  self-orthogonal code then each generator matrix..." $\rightarrow$ 

"If

$C$ is a  self-orthogonal code, then each generator matrix...". 

 

6. Page 4, Paragraph 5: "$\mathbf{F}_{4}$" 

$\rightarrow$ "$\mathbb{F}_{4}$" . Same modifies in other similar situations

 

7. Page 12, Paragraph 10: 

"According to,22 the following quaternary Hermitian LCD codes constucted in

this paper are also optimal..." 

$\rightarrow$ "According to Ref.[22], the following quaternary Hermitian LCD codes constucted in

this paper are also optimal...". 

 

8. Page 12, Paragraph 18:  "the minimum distances of those codes appear very

good in general." make me  confuse. 

$\rightarrow$ "the minimum distances of those codes appear to be the best possible ones ". 

 

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The language is very fluent and professional.

Author Response

Reply to Reviewer 4
       We would like to thank you for    your valuable comments that have been helpful to improve the    paper considerably.   All changes in the manuscript relevant to the comments received  are highlighted in Blue.
       1. {\bf Comment}:  Page 1, line -10 : " Carlet and Guilley in$^{3}$    showed that LCD codes play an important role" $\rightarrow$    "Carlet and Guilley in Ref.$[3]$    showed that LCD codes play an important role".    Modify  in other similar situations
   {\bf Response}:Thanks for your valuable suggestions.       We revised as : "Carlet and Guilley in Ref.$[3]$     showed that LCD codes play an important role.".                2. {\bf Comment}:  Page 2, line -2 : "If there  exists  an $[n,k,d]$ and  $d$ is the largest value present known, then    $\mathcal{C}$ is called a {\it best known }  code and also    denoted $[n,k,d_{o}(n,k)]$." $\rightarrow$ "For a given code $[n,k,d]$,    if $d$ is the largest value present known, then    $\mathcal{C}$ is called a {\it best-known }  code and also    denoted $[n,k,d_{o}(n,k)]$.".
   {\bf Response}:  We   revised as "For a given code $[n,k,d]$,    if $d$ is the largest value present known, then    $\mathcal{C}$ is called a {\it best-known }  code and also    denoted $[n,k,d_{o}(n,k)]$."        3.  {\bf Comment}:  Page 2, line -1 : "If an $\mathcal{C}=[n, k, d_{l}(n,k)]$ LCD    code is" $\rightarrow$    "If a code  $\mathcal{C}=[n, k, d_{l}(n,k)]$ LCD    code is".          {\bf Response}:  We   revised as  "If a code  $\mathcal{C}=[n, k, d_{l}(n,k)]$ LCD    code is''            4. {\bf Comment}: Page 3, line 5:  "$({ u,v})={ uv^{h}}=u_{1}\bar{v_{1}}+u_{2}\bar    {v_{2}}+\cdot\cdot\cdot+u_{n}\bar{v_{n}}.$" $\rightarrow$     "$({ u,v})_{h}={ uv^{2}}=u_{1}\bar{v_{1}}+u_{2}\bar    {v_{2}}+\cdot\cdot\cdot+u_{n}\bar{v_{n}}.$".            {\bf Response}:  We  have  revised  "$({ u,v})_{h}={ uv^{2}}=u_{1}\bar{v_{1}}+u_{2}\bar    {v_{2}}+\cdot\cdot\cdot+u_{n}\bar{v_{n}}.$".        5. {\bf Comment}:  Page 3, line 8: "If    $C$ is a  self-orthogonal code then each generator matrix..." $\rightarrow$    "If   $C$ is a  self-orthogonal code, then each generator matrix...".
   {\bf Response}:  We  have  revised "If   $C$ is a  self-orthogonal code, then each generator matrix...".
       6. {\bf Comment}:  Page 4, Paragraph 5: "$\mathbf{F}_{4}$" $\rightarrow$ "$\mathbb{F}_{4}$" . Same modifies in other similar situations.
   {\bf Response}:  We  have  revised.                7. {\bf Comment}:   Page 12, Paragraph 10:    "According to,22 the following quaternary Hermitian LCD codes constucted in    this paper are also optimal..."    $\rightarrow$ "According to Ref.[22], the following quaternary Hermitian LCD codes constucted in    this paper are also optimal...".
   {\bf Response}: We  have  revised "According to Ref.[22], the following quaternary Hermitian LCD codes constucted in    this paper are also optimal".            8.  {\bf Comment}: Page 12, Paragraph 18:  "the minimum distances of those codes appear very    good in general." make me  confuse.    $\rightarrow$ "the minimum distances of those codes appear to be the best possible ones ".                        {\bf Response}:    We  have revised as "the minimum distances of those codes appear to be the best possible ones ".            

Author Response File: Author Response.pdf