Matrix-Product Codes over Commutative Rings and Constructions Arising from $(\sigma,\delta)$-Codes

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 $R$. 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 generating matrix is given. If $R$ is finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generating matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, results of this paper are used along with previous results of the authors to construct novel MPCs arising from $(\sigma, \delta)$-codes. Some properties of such constructions are also studied.


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][2][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, R denotes a commutative ring with identity and U(R) the multiplicative group of all invertible elements of R. A nonempty subset C of the free R-module R n is called a code over R of length n, and an element of C is called a codeword.If C is an R-submodule of R n , then C is called a linear code over R. e R-submodule of R n generated by a code in R n is obviously a linear code over R, so all codes considered in this paper are linear.If C is a free R-submodule of R n of rank k (i.e., C has an R-basis whose cardinality is k), then C is called a free linear code over R of rank k, and we express this by saying that C is an [n, k]-linear code over R. If C is an [n, k]-linear code over R, we say that a matrix G ∈ M k,n (R) is a generator matrix of C if the rows of G form an R-basis of C. We, thus, write C � xG|x ∈ R k  .On the free R-module R n , consider the (Euclidean) bilinear form 〈•, •〉: R n × R n ⟶ R defined by <(x 1 , . . ., x n ), (y 1 , . . ., y n ) > �  n i�1 x i y i .With respect to this bilinear form, define the dual C ⊥ of any code C over R by C ⊥ � x ∈ R n |〈x, y〉 � 0  for all y ∈ C}.It is easy to check that C ⊥ is a linear code over R if C is so.When C⊆C ⊥ (resp.C � C ⊥ ), we say that C is self-orthogonal (resp., self-dual).A code C is called linear complementary dual (LCD for short) if C ∩ C ⊥ � 0 { } (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 matrixproduct 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 matrixproduct codes over various commutative rings (see, for instance, [6][7][8][9][10][11]).Let C i be an [n, k i ]-linear code over R, for i � 1, . . ., s. Writing codewords of the codes C i in column form, let (c 1 , . . ., c s ) be the n × s matrix whose columns are c 1 ∈ C 1 , . . ., c s ∈ C s .Consider the following subset of the set M n×s (R) of n × s matrices with entries in R: For s ≤ l and a matrix A ∈ M s×l (R), define the matrixproduct code associated with C 1 , . . ., C s and A to be ( As the three R-modules M n×l (R), R nl , and (R n ) l are isomorphic, [C 1 , . . ., C s ]A can be thought of as a code of length nl over R in an obvious way, and we can look at codewords of [C 1 , . . ., C s ]A as elements of either of these three modules.More specifically, if A � (a ij ) and is matrix can be identified with the its corresponding element of R nl , so the codeword (c 1 , . . ., c s )A can be looked at as the following element: On the contrary, as the kth column of the above matrix is  s i�1 a ik c i ∈ R n , the codeword (c 1 , . . ., c s )A can be looked at as the following l-tuple with coordinates from R n : Some of the well-known code constructions turned out to be matrix-product codes.For instance, the Plotkins (u|u + v)−construction and the ternary (u + v + w|2u + v|u)−construction are only examples of matrix-product codes with matrices 1 1 0 1   and , respectively (see [5]).Moreover, recent new constructions of linear codes arising from matrix-product codes have been evolving (see [7,[12][13][14][15][16]).

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 C ⊆ R n 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 matrixproduct code over a commutative ring to be free, and we also give its generator matrix in Corollary 1.
(2) When a code C is free over R, its dual C ⊥ may not be free.Even if C and C ⊥ are both free codes over R of length n, the equality rank(C) + rank(C ⊥ ) � n may not hold.With respect to these issues, it follows from [10, Proposition 2.9] that if R is a finite commutative ring and C is an 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 eorem 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 R, more results are proved.Over such a ring, we generalize in Proposition 3 a well-know fact that tells when the dual of a matrixproduct code is also a matrix-product code. is 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 (σ, δ)-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 matrixproduct codes out of (σ, δ)-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 selfdual.Appropriate highlighting examples are also given throughout.

Matrix-Product Codes over Commutative Rings
In this section, unless further assumptions are stated, R stands for a commutative ring with identity.Proof.
e 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].
is a homomorphism of R-modules.If (c 1 , . . ., c s ) ∈  s i�1 C i is such that ϕ(c 1 , . . ., c s ) � 0, then for each 1 ≤ k ≤ n and 1 ≤ t ≤ l, we have  s j�1 x kj a jt � 0. As A has full rank over R, it follows that for each 1 ≤ k ≤ n and 1 ≤ t ≤ l, we have x kj � 0. erefore ϕ is injective.It is clear, by construction, that ϕ is surjective and, therefore, the rank of Finally, the last statement follows from the bijectivity of ϕ.
Proof.Assume that S � (s 1 , . . ., s k ), and let B � (e 1 , . . ., e k ) be an R-basis of M. Consider the R-module homomorphism us, S is linearly independent and, hence, S is an R-basis of M. □ Remark 1.In contrast with vector spaces over fields, one should be warned that with M as in Lemma 1, a linearly independent system whose cardinality is k is not necessarily an R-basis of M. For instance, looking at Z as a free Z-module of rank 1, we notice that 2 is linearly independent over Z but does not generate Z.
. Let d i be the minimum Hamming distance of C i and D i the minimum Hamming distance of C L i .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 matrixproduct code [C 1 , . . ., C s ]A over a commutative ring when A has full rank.It, further, gives a sufficient condition under which the bound is sharp, generalizing [14, eorem 1].Note that, in the following theorem, we use the multipli- x r ≠ 0, and x i � 0 for r < i ≤ s (so, c � (x 1 , . . ., x r 0, . . ., 0)A); otherwise set r � s.Since 0 ≠ x r ∈ C r , wt(x r ) ≥ d r and, thus, x r has at least d r nonzero components, x i 1 ,r , . . ., x i d r ,r say.Now, for each t � 1, . . ., d r , we have y t � (x i t ,1 , x i t ,2 , . . ., x i t ,s )A ∈ C L r because x j,k � 0 for each j � 1, . . ., l and r < k ≤ s.Since x i t ,r ≠ 0 and A has a full rank over R, we deduce that y t ≠ 0. So, wt(y t ) ≥ D r .Hence, Now assume, further, that C s ⊆C s−1 ⊆ • • • ⊆C 1 and, for every i � 1, . . ., s, there exist On the contrary, as precisely d r components of x r are nonzero and precisely D r components of X r are nonzero, it follows from the definition of the multiplication X r x r that wt(X r x r ) ≤ d r D r .Hence, wt(X r x r ) � d r D r as claimed.

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Remark 2. Note that if R is a field (or even an integral domain), then the requirement on x i and X i in eorem 1 holds automatically.On the contrary, we present here an example which shows that such a requirement is sufficient but not necessary.Let R � Z 4 and consider the matrix ).It can be checked that D 1 � D 2 � 1, the only codeword in C L 1 of weight 1 is (2, 0, 0), and the only codewords in C L 2 of weight 1 are (2, 0, 0) and (0, 0, 2).Set has weight 1.

On the Dual of a Matrix-Product Code over a Finite Commutative Ring
roughout this section, R denotes a finite commutative ring with identity.A nonempty subset of the free R-module M n×m (R) � R nm can be looked at as a code over R of length nm, where a codeword (which is a matrix A ∈ M n×m (R)) is thought of as a word over R of length nm in the obvious way.We consider the following bilinear form on M n×m (R): for A � (a ij ) and B � (b ij ), where B T is the transpose of B and tr(AB T ) is the trace of the n × n matrix AB T .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]).
Proof.Let rank(C j ) � k j for j � 1, . . ., s.Since R is finite, it follows from Lemma 2 that C ⊥ j are free over R of rank n − k j for j � 1, . . ., s. us, by Proposition 2, both From now on, just follow the proof of [2, Lemma 6.1] with the obvious notational adjustments.

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Remark 3. Freeness of the input codes is necessary for the conclusion of Proposition 3 to hold, as the following example shows: let R � Z 20 , C 1 � 10Z 20 , C 2 � 2Z 20 , and and A is nonsingular with (A − 1 ) T � 7 0 0 3  .Now, for Notice that all assumptions of Proposition 3 are satisfied here except that C 1 and C 2 are not free over Z 20 .

respectively, and
Proof.
,n (R) be the respective generator matrices of C ⊥ i , the result now follows from Corollary 1 and Proposition 3.
For r 1 , . . ., r s ∈ R, let Diag(r 1 , . . ., r s ) denote the diagonal matrix of size s × s whose principal-diagonal entry in position i, i is r i for i � 1, . . ., s.
e 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
A ∈ M s×s (R) be such that AA T � Diag(r 1 , . . ., r s ) with r i ∈ U(R) for i � 1, . . ., s, and let C 1 , . . ., C s be linear codes over R of the same length.en Proof.To begin with, as Diag(r 1 , . . ., r s ) is invertible and A is a square matrix over a commutative ring ( [18]), A and A T are invertible too, with By Lemma 3,

Matrix-Product Codes Arising From (σ, δ)-Codes over Finite Commutative Rings
roughout this section, R 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 (σ, δ)-codes over R and, further, give a criterion for self-duality of such codes.We start off by recalling some terminologies and results from [17,19].

(σ, δ)-Codes.
For a ring endomorphism σ of R that maps the identity to itself and a σ-derivation δ of R, let R σ,δ denote the (noncommutative) ring of skew-polynomials  m i�0 a i X i over R with the usual addition of polynomials and multiplication based on the rule is a right divisor in R σ,δ , and let (f) l be the left principal ideal of R σ,δ generated by f. en, R σ,δ /(f) l is both a left R σ,δ -module as well as a free left R-module with an R-basis On the contrary, letting be the companion matrix of f, define the group endomorphism T f : R n ⟶ R n by given by (P(X), (t 0 , . . ., t n−1 ))↦P(T f )(t 0 , . . ., t n−1 ) defines a left action of R σ,δ on R n which makes R n a left R σ,δ -module in an obvious way.Now, the map ϕ f : R n ⟶ R σ,δ /(f) l given by (t 0 , . . . , and we call (t 0 , . . ., t n−1 ) the coordinates of p(X) + (f) l with respect to the basis B. If Note that C consists of the coordinates of all the elements of M. As R is a subring of R σ,δ , M and C are also left R-modules.A linear code C⊆R n is called a principal (f, σ, δ)-code (or just a principal (σ, δ)-code) generated by g if there exist monic skewpolynomials f, g ∈ R σ,δ of degrees n and n − k, respectively, such that g is a right divisor of f in R σ,δ and C � ϕ −1 f ((g) l /(f) l ).Such a code is free over R of rank k (see [5, eorem 1]).A (σ, δ)-code is called a principal (σ, δ)-constacyclic code if it is generated by some monic right divisor of X n − a for some a ∈ U(R).
Starting with a set of monic skew-polynomials g 1 , . . ., g s over R, we give here a construction of a free matrix-product code C over R whose input codes are principal (σ, δ)-codes generated by the g i 's and, further, give its generator matrix in terms of the matrix of the code and the coefficients of the g i 's.We also give a construction of the dual C ⊥ of C under certain extra assumptions and give its generator matrix (a parity-check matrix of C ).
For every j � 1, . . ., s, let σ j be a ring endomorphism of R that maps the identity to itself, δ j a σ j -derivation of R, g j (X) �  n−k j i�0 g i,j X i ∈ R σ j ,δ j monic, and C j the principal (σ j , δ j )-code over R generated by g j (X) (so, there exists a monic f j (X) ∈ R σ j ,δ j of degree n of which g j (X) is a right divisor in R σ j ,δ j ).By [3, eorem 2.7], a generator matrix G j ∈ M k j ×n (R) of C j is given by 6 Journal of Mathematics G j � g 0,j . . .g n−k j ,j 0 0 . . .0 g (1)   0,j . . .g (1)  n−k j ,j σ j g n−k j ,j   0 . . .0 g (2)   0,j . . .g (2)   n−k j ,j g (2)   n−k j +1,j σ 2 j g n−k j ,j   . . .0 e matrices G j take more elegant shapes if δ j � 0 where, by [3, Corollary 2.8], we would have On the contrary, if further σ j are ring automorphisms of R and g j (X) are also left divisors in R σ j ,δ j of f j (X) �  n i�0 a i,j X i for all j � 1, . . ., s with where 0,j � δ j (h (i−1) 0,j ) − a 0,j σ j (h (i−1) n−1,j ) (ii) h (i)  t,j � δ j (h (i−1) t,j ) + σ j (h (i−1) t−1,j ) − a t,j σ j (h (i− is a generator matrix for the matrix-product code [g 1 , . . ., g s ]A. Proof.By [5, eorem 1] (see also [17]), C i is free of rank k i for every i � 1, . . ., s.Now, applying Corollary 1 yields the claimed conclusion.

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Proposition 5.Besides the assumptions of Proposition 4, assume further that σ i is a ring automorphism of R, g i (X) is also a left divisor of f i (X) for i � 1, . . ., s, and given by

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Example 1.Let R be finite of characteristic 2, α, β ∈ R with α 2 + α + 1 � 0, and σ 1 , σ 2 ring automorphisms of R with σ 1 (α) � α 2 and σ 2 (β) � β.We present several principal σ i -codes of length 4 and use them to construct many matrixproduct codes. Step Let C 1 and C 2 be the principal σ 1 -codes of length 4 over R generated, respectively, by g 1 and h 1 .By Section 4.1, is a generator matrix of C 1 , and is a generator matrix of C 2 .On the contrary, is a generator matrix of C ⊥ 1 , and 2,1 Step Let C 3 and C 4 be the principal σ 2 -codes of length 4 over R generated, respectively, by g 2 and h 2 .By Section 4.1, is a generator matrix of C 3 , and is a generator matrix of C 4 .On the contrary, is a generator matrix of C ⊥ 3 , and 1,2 1,2 2,2 R) 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 σ i -codes such as [g i g j ]A and Journal of Mathematics we can construct generator matrices of different combinations of the codes C i and their dual codes for i � 1, 2, 3, 4.
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 A, B, and D are square and nonsingular.For instance, following Remark 3, let R be Z 20 and A � 3 0 0 7  .
en A is nonsingular and (A − 1 ) T � 7 0 0 3  .As in Step 3, a generator matrix of By Proposition 3, a generator matrix of ([g Note that Besides the assumptions of Proposition 5, let us now assume further that, for every j � 1, . . ., s, δ j � 0, g 0,j ∈ U(R), k � k 1 � k 2 � • • • � k s with n � 2k, and denote R σ j ,0 by R σ j .Proposition 6. Keep the assumptions as above.Assume further that A � (a ij ) ∈ M s×s (R) is such that AA T � Diag(r 1 , . . ., r s ) with r 1 , . . ., r s ∈ U(R) and that, for every j � 1, . . ., s, either of the following statements holds: (i) g j (X) is a right divisor in R σ j of X n − a j for some a j ∈ U(R), C j is the principal (X n − a j , σ j )-constacyclic code generated by g j (X), and σ k j j (h −1 0,j )h * j (X) � g j (X), where g j (X)h j (X) � X n −σ −k j j (a j ).(ii) For any l j ∈ 0, . . ., k j  ,  l j i�0 σ k j −1 j (g i,j )g i+k j −l j ,j � 0.
Proof.By [3, Corollary 3.7], the statements (i) and (ii) are equivalent and, further, they are equivalent to the condition of C j being self-dual.Now, apply Corollary 1 to get that [g 1 , . . ., g s ]A is self-dual.
Keep the notations and assumptions of Section 4.1.ForA ∈ M s×l (R), we denote the matrix-product code[C 1 , . .., C s ]A by [g 1 , .. ., g s ]A in order to emphasize a way of constructing a free matrix-product over R out of a wellchosen set of skew-polynomials over R, as the following results indicate.
29)is a generator matrix for the dual matrix-product code([g 1 , . .., g s ]A) ⊥ .Proof.By the presentation in Subsection 4.1, H i is a generator matrix of C ⊥ i for i � 1, . . ., s.Now, apply Corollary 2 to get the desired conclusion.