Minimal State-Space Representation of Convolutional Product Codes

Round 1

Reviewer 1 Report

In this paper, the authors study a state-space representation of product convolutional codes. It is shown how a minimal state-space representation of a product convolutional code can be obtained from minimal state-space representations of its component codes.

Remarks.

1. The authors should better motivate their research topic. Theorem 5 in [4] proves that if generator matrices of component codes are minimal basic encoding matrices, the generator matrix of the corresponding convolutional code is a minimal basic encoding matrix. The main result of the current paper straightforwardly generalizes this conclusion by considering state-space representations of product codes. Since matrices A and B determine state diagram transitions,  these matrices of the minimal state-space representation can be straightforwardly constructed from the corresponding matrices of component codes having minimal state-space representation. 
2. The main idea behind product codes is to decode a complex code by applying iterative decoding of much more simple component codes. State-space representations of product codes are not used in practical decoding schemes.   

 In my opinion, the subject of this paper can be interesting to a limited audience; both the theoretical and practical contributions of this paper are rather marginal.

Author Response

We thank this Reviewer for the comments and remarks.    Regarding the remarks 1 and 2, we have made an effort to better clarify the main goal of the paper in the introduction. In fact, this goal is mainly theoretical rather than practical, as most of the results of input state output (iso) representations of convolutional codes. These representations provide, in our opinion, a nice and natural framework to work with convolutional codes, but their direct practical use is questionable so far. For example, to the best of our knowledge, there is no better construction or decoding algorithm within this framework than using standard  representations. As this is a mathematical journal we hope and believe these results fit well in this journal.    The main result of the paper is to show how one can build an iso representation of the product code directly from the iso representations of the component convolutional codes (without passing to the encoding matrices). We show that this can be done and provide an algorithm for doing so. This is not straightforward.  We do hope that these facts are now properly explained in this new version.   

Reviewer 2 Report

The manuscript mathematics-1185775 "Minimal State-Space Representation of Convolutional Product Codes" by Joan-Josep Climent, Diego Napp, Raquel Pinto, and Verónica Requena presents new results on two-dimensional product convolutional codes. Namely, the authors describe an algorithm for deriving state-space representations of a given code from the basis vectors of corresponding horizontal and vertical codes. In particular, they give a systematic procedure for building such a representation with minimal dimension. In my opinion the manuscript is clearly written, and after working through the text, I did not find any issues with equations or theorems. The field of this work is coding theory; I cannot comment how well it fits the subject area of the Journal. Otherwise, I am happy to recommend this work for publication. My only optional suggestion to the authors: it would be nice to explain the importance of notions of reachability and observability, outside of their use in the proofs.

Author Response

We thank this Reviewer for the positive comments on our work. Following the suggestion we have added an explanation of the meaning and importance of the notions of observability and reachability in page 4 (from line 118 to 124). We do hope that this will help to better understand these important notions. 

Reviewer 3 Report

In this paper the authors presented a constructive methodology to build a minimal state-space representation $(A, B, C, D)$ of a convolutional product code from two minimal state-space representations, $(A_h, B_h, C_h, D_h)$ and $(A_v, B_v, C_v, D_v)$ of an horizontal and a vertical convolutional code, respectively. 

I think that in Theorem 3 the notation $v_{\ell}$ (which introduced in Theorem 7) can be introduced. This way the next description will become even clearer and easier to read.

Since PID (p.3., r.2, after 87 row) use only once, I recommend replaced with the full name, i.e. Principle Ideal Domain.

193 row: Now, as a consequence of theorems 6 and 7 ... replace with

Now, as a consequence of Theorems 6 and 7 ... 

After minor revision, I recommend the acceptance and publication of this article.

Author Response

We thank this Reviewer for the positive comments on our work. Following the suggestions we have modified all the considerations required.

Round 2

Reviewer 1 Report

  I did not change the opinion concerning this paper.  I would not argue if the authors would obtain similar results for more general linear system but they focused on a particular code construction.    

1. In my opinion, the attraction of new mathematical tools to investigating this code construction should be justified by proving new properties of the code construction which could not be obtained by using mathematical methods commonly  used in the field of coding. For example in [29,47], the use of  state-space representations of convolutional codes made it possible to introduce new algebraic constructions of convolutional codes.
2. The main idea behind product codes is to decode a complex code by applying iterative decoding of much more simple component codes. State-space representations of product codes are not used in practical decoding schemes. The authors should better motivate their research topic. What  coding or maybe code identification problem can be solved by applying the results of this paper? 

Author Response

We thank this Reviewer again for his/her comments. We have tried our best to further motivate the importance of the results of this paper in this new revised version.

This can be found  in the section "Introduction". From line42 to 69, we motivate better our research topic by presenting some possible applications that can be derived from our results. In particular, we point out that there have been new advances in algebraic decoding algorithms of convolutional codes using state space representations that could be used to simplify the decoding of product convolucional codes given in terms of state space representations.

We do hope that now this argument will justify enough the publication of these results that, in our opinion, have most of their value in the area of realization theory. 

Meanwhile, we would not argue if this was an engineer journal where there must be a direct application of the results presented. However, this is a mathematical journal and the results we derived in this work are mainly theoretical. For this reason, we do not think it is resaonable to reject this paper for lack of straightforward applications. The mathematical relevance and correctness of the results have been backed by the other reviews.