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Boulagouaz, Mhammed; Deajim, Abdulaziz

doi:https://doi.org/10.1155/2021/5521067

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Research Article | Open Access

Volume 2021 | Article ID 5521067 | https://doi.org/10.1155/2021/5521067

Matrix-Product Codes over Commutative Rings and Constructions Arising from -Codes

Mhammed Boulagouaz¹and Abdulaziz Deajim²

Academic Editor: Marco Fontana

Received28 Jan 2021

Accepted11 Feb 2021

Published03 Mar 2021

Abstract

A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring . A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is free when its input codes are all free, in which case a generator matrix is given. If is finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generator matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, the results of this paper are used along with previous results of the authors to construct novel MPCs arising from -codes. Some properties of such constructions are also studied.

1. Introduction

Over the past two decades, studying codes over commutative rings (especially finite ones) and their properties has been attracting a great deal of attention. As far as engineering applications are concerned, it is understood that codes over special types of commutative rings are more relevant; namely, finite Frobenius rings. For a very nice survey of such development, and for relevant references, one can check [1–3]. From a purely mathematical point of view, however, the authors believe that when certain results can be verified over more general commutative rings, then it would be reasonable to pursue such approach for the sake of mathematical interest (an example of such approach is seen in [3]).

1.1. Linear Codes

In this work, if not otherwise specified, denotes a commutative ring with identity and the multiplicative group of all invertible elements of . A nonempty subset of the free -module is called a code over of length , and an element of is called a codeword. If is an -submodule of , then is called a linear code over . The -submodule of generated by a code in is obviously a linear code over , so all codes considered in this paper are linear. If is a free -submodule of of rank (i.e., has an -basis whose cardinality is ), then is called a free linear code over of rank , and we express this by saying that is an -linear code over . If is an -linear code over , we say that a matrix is a generator matrix of if the rows of form an -basis of . We, thus, write .

On the free -module , consider the (Euclidean) bilinear form defined by . With respect to this bilinear form, define the dual of any code over by for all . It is easy to check that is a linear code over if is so. When (resp. ), we say that is self-orthogonal (resp., self-dual). A code is called linear complementary dual (LCD for short) if (see [4]).

1.2. Matrix-Product Codes

Constructing codes (with certain properties) from smaller ones has lately been an important problem in pursuit by coding theorists. In particular, studying the effect of the properties of the smaller codes on the properties of the constructed codes is crucial. An interesting such construction is the construction of matrix-product codes over finite fields first introduced by Blackmore and Norton in [5]. Ever since the introduction, researchers have been trying to pave new tracks in this area. Particularly relevant to us here are constructions of matrix-product codes over various commutative rings (see, for instance, [6–11]).

Let be an -linear code over , for . Writing codewords of the codes in column form, let be the matrix whose columns are . Consider the following subset of the set of matrices with entries in :

For and a matrix , define the matrix-product code associated with and to be

As the three -modules , , and are isomorphic, can be thought of as a code of length over in an obvious way, and we can look at codewords of as elements of either of these three modules. More specifically, if and for , then the codeword of is the matrix:

This matrix can be identified with the its corresponding element of , so the codeword can be looked at as the following element:

On the contrary, as the th column of the above matrix is , the codeword can be looked at as the following -tuple with coordinates from :

It should be noted that is linear if all are linear.

Some of the well-known code constructions turned out to be matrix-product codes. For instance, the Plotkins construction and the ternary construction are only examples of matrix-product codes with matrices and , respectively (see [5]). Moreover, recent new constructions of linear codes arising from matrix-product codes have been evolving (see [7, 12–16]).

1.3. Points of Investigation and Contributions

In general, some of the serious differences between linear codes over fields versus linear codes over commutative rings are apparent from the following:(1)A linear code may not be free. Due to this, it is not possible to talk about a generator matrix of a nonfree code in the sense of the definition of such a matrix we have given. In this regard, we give in Proposition 2 sufficient conditions for a matrix-product code over a commutative ring to be free, and we also give its generator matrix in Corollary 1.(2)When a code is free over , its dual may not be free. Even if and are both free codes over of length , the equality may not hold. With respect to these issues, it follows from [10, Proposition 2.9] that if is a finite commutative ring and is an -linear code over , then is an -linear code over . So, in our relevant results, we work over finite commutative rings. Nonetheless, as was stated earlier, whenever a result can be proved valid over more general rings, then we state it and prove in such generality.(3)In an effort to extend results on the minimum Hamming distance of matrix-product codes over finite fields to those over general commutative rings, we prove in Theorem 1 that a well-known lower bound for the minimum distance of a matrix-product code over a finite field or a finite chain ring remains valid over a commutative ring and we, further, give a sufficient condition for such a lower bound to be sharp.(4)When we impose finiteness on , more results are proved. Over such a ring, we generalize in Proposition 3 a well-know fact that tells when the dual of a matrix-product code is also a matrix-product code. This is used in Corollary 2 to give a generator matrix of the dual for a matrix-product code, and it is also used in Corollary 3 to give characterizations of self-dual, self-orthogonal, and LCD matrix-product codes.(5)As an interesting application, we study in Section 4 matrix-product codes arising from -codes over finite commutative rings. In this section, we bring together results from the authors’ work [17] and results proved in this paper to construct matrix-product codes out of -codes, give generator matrices for such codes and their dual codes (Propositions 4 and 5), and give a criterion in Proposition 6 which tests when such a code is self-dual. Appropriate highlighting examples are also given throughout.

2. Matrix-Product Codes over Commutative Rings

In this section, unless further assumptions are stated, stands for a commutative ring with identity.

Definition 1. Let with .(i)If the rows of are linearly independent over , we say that has full rank over .(ii)If there is such that (the identity matrix), we say that is right invertible and is the right inverse of . Left invertibility is defined similarly. If , we say that is invertible if it has both right and left inverses.(iii)If and the determinant is a unit of , then we say that is nonsingular.

Proposition 1. If , then the following statements are equivalent:(i)A is invertible.(iiA is nonsingular. If, further, is finite, then the above two statements are equivalent to the following:(iii)A has full rank.

Proof. The equivalence of the first two statements follows from the standard argument of computing the inverse of a square matrix ([18]). For the last statement, see [10, Corollary 2.8].

Proposition 2. Let be of full rank and be -linear codes over for . Then the matrix-product code is an -linear code over . If, further, is finite, then

Proof. The mapis a homomorphism of -modules. If is such that , then for each and , we have . As has full rank over , it follows that for each and , we have . Therefore is injective. It is clear, by construction, that is surjective and, therefore, the rank of is equal to the rank of , which is . Finally, the last statement follows from the bijectivity of .

Lemma 1. Let be a free -module of rank , and a system of elements of . If generates over , then is a basis of over .

Proof. Assume that , and let be an -basis of . Consider the -module homomorphismIf with , then . Since are linearly independent over , . Thus, is linearly independent and, hence, is an -basis of .

Remark 1. In contrast with vector spaces over fields, one should be warned that with as in Lemma 1, a linearly independent system whose cardinality is is not necessarily an -basis of . For instance, looking at as a free -module of rank 1, we notice that 2 is linearly independent over but does not generate .

Corollary 1. Let be of full rank, and be -linear codes over , respectively. If is a generator matrix of for , respectively, then is free with the following generator matrix :

Proof. By Proposition 2, is free of rank over . The set consisting of the rows of the matrix is clearly a generating system of the code over . Since the cardinality of is equal to the rank of , it follows from Lemma 1 that is a basis of over . Hence, is a generator matrix of .
Let . For , let be the th row of and the left -submodule of generated by (so, ). Let be the minimum Hamming distance of and the minimum Hamming distance of . Generalizing its counterparts over a finite field ([5]) and a finite chain ring ([6]), the theorem below gives a lower bound for the minimum Hamming distance of a matrix-product code over a commutative ring when has full rank. It, further, gives a sufficient condition under which the bound is sharp, generalizing [14, Theorem 1]. Note that, in the following theorem, we use the multiplication map defined byfor and .

Theorem 1. Keep the above notation. If is of full rank, then the minimum distance of the matrix-product code satisfies the following inequality:If, furthermore, and, for every , there exist and such that , , and , then

Proof. Let . There exist some , , and for (so, ); otherwise set . Since , and, thus, has at least nonzero components, say. Now, for each , we have because for each and . Since and has a full rank over , we deduce that . So, . Hence,Now assume, further, that and, for every , there exist and such that , , and . By the first part of this proof, we have . Let be such that . Take and so that , , and . We show that and , which settles the proof. Write and for . Set for with for . As , for all . Also, for all . Thus, andAs , . On the contrary, as precisely components of are nonzero and precisely components of are nonzero, it follows from the definition of the multiplication that . Hence, as claimed.

Remark 2. Note that if is a field (or even an integral domain), then the requirement on and in Theorem 1 holds automatically. On the contrary, we present here an example which shows that such a requirement is sufficient but not necessary. Let and consider the matrix . It is clear that is of full rank. Consider and . It can be checked that , the only codeword in of weight 1 is , and the only codewords in of weight 1 are and . Set and , so and . It is easily seen that . Nonetheless, since the codewordhas weight 1.

3. On the Dual of a Matrix-Product Code over a Finite Commutative Ring

Throughout this section, denotes a finite commutative ring with identity. A nonempty subset of the free -module can be looked at as a code over of length , where a codeword (which is a matrix ) is thought of as a word over of length in the obvious way. We consider the following bilinear form on :for and , where is the transpose of and is the trace of the matrix .

Our next goal is to give sufficient conditions for the dual of a matrix-product code to also be a matrix-product code, generalizing similar results over finite fields and finite chain rings (see [5, 6]).

Lemma 2. (see [10, Proposition 2.9]). If is an -linear code over , then is an -linear code over .

Proposition 3. If is nonsingular, and are free linear codes over of length , then .

Proof. Let for . Since is finite, it follows from Lemma 2 that are free over of rank for . Thus, by Proposition 2, both and are free withFrom now on, just follow the proof of [2, Lemma 6.1] with the obvious notational adjustments.

Remark 3. Freeness of the input codes is necessary for the conclusion of Proposition 3 to hold, as the following example shows: let , , , and . It can be easily seen that , , and is nonsingular with . Now, for and , it is clear that in . So, . Notice that all assumptions of Proposition 3 are satisfied here except that and are not free over .

Corollary 2. Keep the assumptions of Proposition 3. If are generator matrices of , respectively, and , then a generator matrix of is

Proof. If is of rank over for , it follows from Lemma 2 that is free of rank for . Letting be the respective generator matrices of , the result now follows from Corollary 1 and Proposition 3.
For , let denote the diagonal matrix of size whose principal-diagonal entry in position is for .

Lemma 3. If are linear codes over of the same length, , and for some . Then, .

Proof. As , . Since is a unit, for , and thus the claimed conclusion follows immediately.
The following result gives characterizations of self-dual, self-orthogonal, and LCD matrix-product codes over finite commutative rings (for part 3, see [8] as well).

Corollary 3. Let be such that with for , and let be linear codes over of the same length. Then(1)is self-dual if and only ifis self-dual for every.(2)is self-orthogonal if and only ifis self-orthogonal for every.(3)is LCD if and only ifis LCD for every.

Proof. To begin with, as is invertible and is a square matrix over a commutative ring ([18]), and are invertible too, withBy Lemma 3,Also, by Proposition 3,(1)By (19) and (20),(2)Proving the self-orthogonality statement is similar.(3)We have

4. Matrix-Product Codes Arising From -Codes over Finite Commutative Rings

Throughout this section, denotes a finite commutative ring with identity. We use here some results from the authors’ paper [17] combined with results from the previous sections to construct matrix-product codes based on -codes over and, further, give a criterion for self-duality of such codes. We start off by recalling some terminologies and results from [17, 19].

4.1. -Codes

For a ring endomorphism of that maps the identity to itself and a -derivation of , let denote the (noncommutative) ring of skew-polynomials over with the usual addition of polynomials and multiplication based on the rule for . Let be a monic skew-polynomial, for some . Fix any monic of degree of which is a right divisor in , and let be the left principal ideal of generated by . Then, is both a left -module as well as a free left -module with an -basis . On the contrary, lettingbe the companion matrix of , define the group endomorphism by

Then the map given by defines a left action of on which makes a left -module in an obvious way. Now, the map given by is a left -module isomorphism. For every , there is a unique (of degree at most ) such that . We can see that , and we call the coordinates of with respect to the basis .

If is a left -submodule of , the left -submodule of is called an -code (or just a -code) of length over . Note that consists of the coordinates of all the elements of . As is a subring of , and are also left -modules. A linear code is called a principal -code (or just a principal -code) generated by if there exist monic skew-polynomials of degrees and , respectively, such that is a right divisor of in and . Such a code is free over of rank (see [5, Theorem 1]). A -code is called a principal -constacyclic code if it is generated by some monic right divisor of for some .

Starting with a set of monic skew-polynomials over , we give here a construction of a free matrix-product code over whose input codes are principal -codes generated by the ’s and, further, give its generator matrix in terms of the matrix of the code and the coefficients of the ’s. We also give a construction of the dual of under certain extra assumptions and give its generator matrix (a parity-check matrix of ).

For every , let be a ring endomorphism of that maps the identity to itself, a -derivation of , monic, and the principal -code over generated by (so, there exists a monic of degree of which is a right divisor in ). By [3, Theorem 2.7], a generator matrix of is given bywhere(1) for (2) for (3) for and

The matrices take more elegant shapes if where, by [3, Corollary 2.8], we would have

On the contrary, if further are ring automorphisms of and are also left divisors in of for all with for , then (by [3, Theorem 5.1]) a generator matrix of , for , iswhere(1) for (2)for and (i)(ii)(3)for and (i)(ii)

4.2. Matrix-Product Codes Arising from -Codes

Keep the notations and assumptions of Section 4.1. For , we denote the matrix-product code by in order to emphasize a way of constructing a free matrix-product over out of a well-chosen set of skew-polynomials over , as the following results indicate.

Proposition 4. Keep the notations and assumptions of Section 4.1. Let be of full rank. Then, the matrix given byis a generator matrix for the matrix-product code .

Proof. By [5, Theorem 1] (see also [17]), is free of rank for every . Now, applying Corollary 1 yields the claimed conclusion.

Proposition 5. Besides the assumptions of Proposition 4, assume further that is a ring automorphism of , is also a left divisor of for , and is nonsingular with . Then the matrix given byis a generator matrix for the dual matrix-product code .

Proof. By the presentation in Subsection 4.1, is a generator matrix of for . Now, apply Corollary 2 to get the desired conclusion.

Example 1. Let be finite of characteristic 2, with , and ring automorphisms of with and . We present several principal -codes of length 4 and use them to construct many matrix-product codes.

Step 1. Consider , with , , and . It can be checked thatLet and be the principal -codes of length 4 over generated, respectively, by and . By Section 4.1,is a generator matrix of , andis a generator matrix of . On the contrary,is a generator matrix of , andis a generator matrix of . Notice that .

Step 2. Consider , with , , and . It can be checked thatLet and be the principal -codes of length 4 over generated, respectively, by and . By Section 4.1,is a generator matrix of , andis a generator matrix of . On the contrary,is a generator matrix of , andis a generator matrix of . Notice that and .

Step 3. Let be all full-rank matrices. By Proposition 4, we can easily construct the generator matrices of many matrix-product codes out of different combinations of the above principal -codes such as and , , , , and for . For instance, is a generator matrix of , is a generator matrix of , and is a generator matrix of . In a similar manner, we can construct generator matrices of different combinations of the codes and their dual codes for .

Step 4. Utilizing Proposition 3, we can give the generator matrices of the dual codes of all of the above matrix-product codes when the matrices , , and are square and nonsingular. For instance, following Remark 3, let be and . Then is nonsingular and . As in Step 3, a generator matrix of isBy Proposition 3, a generator matrix of isNote thatBesides the assumptions of Proposition 5, let us now assume further that, for every , , , with , and denote by .

Proposition 6. Keep the assumptions as above. Assume further that is such that with and that, for every , either of the following statements holds:(i)is a right divisor inoffor some,is the principal-constacyclic code generated by, and, where.(ii)For any,.Then, the matrix-product code is self-dual.

Proof. By [3, Corollary 3.7], the statements (i) and (ii) are equivalent and, further, they are equivalent to the condition of being self-dual. Now, apply Corollary 1 to get that is self-dual.

Example 2. Let , , and . Note that is a ring automorphism of of order 2, , and is a unit in of order 2. Set . Then and . Let . So, . Using the multiplication rule in , we haveThus, satisfies condition (2) of Proposition 6 and, hence, it generates a self-dual -constacyclic code of length 4 over . It follows from the paragraph following Proposition 4 that a generator matrix of is . Given the matrix , which is orthogonal and, thus, quasiorthogonal, we conclude from Propositions 6 and 4, respectively, that the matrix-product code is self-dual with the following generator matrix:

Data Availability

All data are contained in the manuscript.

Disclosure

An earlier version of this manuscript was posted on arXiv (see [20]).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors deeply thank the many colleagues and experts whose valuable comments helped them in improving the paper.

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Copyright

Copyright © 2021 Mhammed Boulagouaz and Abdulaziz Deajim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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