# Characterizations and Properties of Monic Principal Skew Codes over Rings

Authors: Mhammed Boulagouaz; Abdulaziz Deajim
Journal: Security and Communication Networks
 (2022)

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## Abstract

LetA be a ring with identity, σ a ring endomorphism of A that maps the identity to itself, δ a σ-derivation of A, and consider the skew-polynomial ring AX;σ,δ. When A is a finite field, a Galois ring, or a general ring, some fairly recent literature used AX;σ,δ to construct new interesting codes (e.g., skew-cyclic and skew-constacyclic codes) that generalize their classical counterparts over finite fields (e.g., cyclic and constacyclic linear codes). This paper presents results concerning monic principal skew codes, called herein monic principalf,σ,δ-codes, where f∈AX;σ,δ is monic. We provide recursive formulas that compute the entries of both a generator matrix and a control matrix of such a code C. When A is a finite commutative ring and σ is a ring automorphism of A, we also give recursive formulas for the entries of a parity-check matrix of C. Also, in this case, with δ=0, we present a characterization of monic principal σ-codes whose dual codes are also monic principal σ-codes, and we deduce a characterization of self-dual monic principal σ-codes. Some corollaries concerning monic principal σ-constacyclic codes are also given, and a good number of highlighting examples is provided.


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## Body

## 1. Introduction

### 1.1. State of the Art

LetA be a ring with identity, σ a ring endomorphism of A that maps the identity to itself, and δ a σ-derivation of A (i.e., δa+b=δa+δb and δab=σaδb+δab for all a,b∈A). Denote by Aσ,δ the skew-polynomial ring(1)AX;σ,δ=∑i=0n−1aiXi|n∈ℕ,ai∈A.Recall thatAσ,δ has the same additive-group structure as that of the usual ring of polynomials AX but has multiplication twisted based on the rule Xa=σaX+δa for a∈A and extended associatively and distributively to all elements of Aσ,δ. This obviously makes Aσ,δ a noncommutative ring unless δ=0, σ is the identity, and A is commutative (in which case Aσ,δ is nothing but AX). In case δ=0, we use the notation Aσ instead of Aσ,0.For a finite fieldF and a ring automorphism σ of F, Boucher et al. [1] used Fσ to introduce the notion of a skew-cyclic code C over F of length n as a code satisfying σan−1,σa0,σa1,…,σan−2∈C for any a0,a1,…,an−2,an−1∈C. This is obviously a generalization of the classical notion of cyclic codes over finite fields (when σ is the identity). It is also shown therein that the class of skew-cyclic codes over finite fields gives a supply of codes with good coding and decoding properties (see also [2–5]). When a monic f∈Fσ generates a two-sided ideal in Fσ, then, Fσ/f is a (noncommutative) principal left-ideal ring. In particular, when the order of σ divides n, then, Xn−1 is a two-sided ideal in Fσ (see [1]). When, further, g∈Fσ is a right divisor of Xn−1, Boucher et al. [1] studied the skew-cyclic code generated by g, which is associated with the principal left ideal g/Xn−1 of Fσ/Xn−1. The structure of such an ideal puts some restrictions on the code (for instance, Xn−1 must be a two-sided ideal, which is ensured by some arithmetical condition on n).To further generalize the notion of skew-cyclic codes, Boucher and Ulmer [2] introduced codes defined as modules over Fσ. Among other things, this new construction has the advantage of removing some of the constraints on the lengths of skew-cyclic codes alluded to above. Boucher et al. [6] relaxed the requirement on the field of coefficients by considering skew-polynomial rings over Galois rings enabling further generalizations and improvements (see also [7]). Boulagouaz and Leroy [8] took this generalization further by letting the ring of coefficients be any ring A with σ a ring endomorphism of A and δ a σ-derivation of A. A nice recent generalization in a different direction may be found in [9]. For other references on skew codes over rings, see [10, 11]; and for more references, see ([12], Chapter 6) and ([13], Chapter 11).


### 1.2. Contributions and Map of the Article

(i)
For a ringA (not necessarily finite nor commutative), an endomorphism σ of A, and a σ-derivation δ of A, the following is performed:(1)
In Section2, we revisit the main definitions of [8] and, particularly, make precise the notions of monic principal f,σ,δ-codes, σ-codes, f,σ,δ-constacyclic codes, and σ-constacyclic codes over A.(2)
Section3 aims mainly at improving ([8], Theorem 1) computationally by giving a generator matrix of a monic principal f,σ,δ-code (resp. a monic principal σ-code) over A using recursive formulas introduced by means of a list of lemmas; see Theorem 1 (resp. Corollary 5).(3)
In Section4, we present precise and more practical recursive formulas which yield, in Theorem 2, the entries of a control matrix of a monic principal f,σ,δ-code C over A whose generating polynomial is both a right and left divisor of f. This gives Theorem 2 a practicality advantage over ([8], Corollary 1). Furthermore, for a monic principal σ-code (resp. a monic principal σ-constacyclic code) over A, the control matrix given in Theorem 2 takes a better shape; see Corollary 6 (resp. Corollary 7).(ii)
For a finite commutative ringA and an automorphism σ of A, the following is performed:(1)
In Section5, we characterize the monic principal σ-codes over A whose dual codes are also monic principal σ-codes, strengthening and extending ([3], Theorem 1); see Theorem 3 and the paragraph that precedes it. Consequently, we give in Corollary 8 a generator matrix of the dual of a monic principal σ-constacyclic code over A, and we further introduce, in Corollary 9, a characterization of self-dual monic principal σ-codes over A in such a way that generalizes and strengthens ([2], Corollary 4).(2)
In Section6, we begin by introducing the notion of a parity-check matrix of a free f,σ,δ-code over a general ring with an endomorphism σ. We then go back to the assumptions on the ring A being finite and commutative and σ an automorphism of A, where we construct a parity-check matrix of a monic principal f,σ,δ-code C over A showing also how to extract such a matrix from a control matrix of C; see Theorem 4. Furthermore, for a monic principal σ-code (resp. a monic principal σ-constacyclic code) over A, the parity-check matrix given in Theorem 4 takes a better shape; see Corollary 10 (resp. Corollary 11). On the other hand, with the crucial assumption that σ is an automorphism of A, we show in Corollary 12 that the parity-check matrix given in Corollary 11 can be obtained without the assumption that the monic principal σ-constacyclic code is generated by some monic g∈Aσ that is also a left divisor of Xn−a.(iii)
Throughout the article, a good number of highlighting examples is given. An earlier preprint of this article is in reference [14]. Some results from this article were used in [15] to construct novel matrix-product codes arising from σ,δ-codes. Other applications of skew codes over rings can be found in ([12], Chapter 6) and ([13], Chapter 11) and the references therein.


## 1.1. State of the Art

LetA be a ring with identity, σ a ring endomorphism of A that maps the identity to itself, and δ a σ-derivation of A (i.e., δa+b=δa+δb and δab=σaδb+δab for all a,b∈A). Denote by Aσ,δ the skew-polynomial ring(1)AX;σ,δ=∑i=0n−1aiXi|n∈ℕ,ai∈A.Recall thatAσ,δ has the same additive-group structure as that of the usual ring of polynomials AX but has multiplication twisted based on the rule Xa=σaX+δa for a∈A and extended associatively and distributively to all elements of Aσ,δ. This obviously makes Aσ,δ a noncommutative ring unless δ=0, σ is the identity, and A is commutative (in which case Aσ,δ is nothing but AX). In case δ=0, we use the notation Aσ instead of Aσ,0.For a finite fieldF and a ring automorphism σ of F, Boucher et al. [1] used Fσ to introduce the notion of a skew-cyclic code C over F of length n as a code satisfying σan−1,σa0,σa1,…,σan−2∈C for any a0,a1,…,an−2,an−1∈C. This is obviously a generalization of the classical notion of cyclic codes over finite fields (when σ is the identity). It is also shown therein that the class of skew-cyclic codes over finite fields gives a supply of codes with good coding and decoding properties (see also [2–5]). When a monic f∈Fσ generates a two-sided ideal in Fσ, then, Fσ/f is a (noncommutative) principal left-ideal ring. In particular, when the order of σ divides n, then, Xn−1 is a two-sided ideal in Fσ (see [1]). When, further, g∈Fσ is a right divisor of Xn−1, Boucher et al. [1] studied the skew-cyclic code generated by g, which is associated with the principal left ideal g/Xn−1 of Fσ/Xn−1. The structure of such an ideal puts some restrictions on the code (for instance, Xn−1 must be a two-sided ideal, which is ensured by some arithmetical condition on n).To further generalize the notion of skew-cyclic codes, Boucher and Ulmer [2] introduced codes defined as modules over Fσ. Among other things, this new construction has the advantage of removing some of the constraints on the lengths of skew-cyclic codes alluded to above. Boucher et al. [6] relaxed the requirement on the field of coefficients by considering skew-polynomial rings over Galois rings enabling further generalizations and improvements (see also [7]). Boulagouaz and Leroy [8] took this generalization further by letting the ring of coefficients be any ring A with σ a ring endomorphism of A and δ a σ-derivation of A. A nice recent generalization in a different direction may be found in [9]. For other references on skew codes over rings, see [10, 11]; and for more references, see ([12], Chapter 6) and ([13], Chapter 11).


## 1.2. Contributions and Map of the Article

(i)
For a ringA (not necessarily finite nor commutative), an endomorphism σ of A, and a σ-derivation δ of A, the following is performed:(1)
In Section2, we revisit the main definitions of [8] and, particularly, make precise the notions of monic principal f,σ,δ-codes, σ-codes, f,σ,δ-constacyclic codes, and σ-constacyclic codes over A.(2)
Section3 aims mainly at improving ([8], Theorem 1) computationally by giving a generator matrix of a monic principal f,σ,δ-code (resp. a monic principal σ-code) over A using recursive formulas introduced by means of a list of lemmas; see Theorem 1 (resp. Corollary 5).(3)
In Section4, we present precise and more practical recursive formulas which yield, in Theorem 2, the entries of a control matrix of a monic principal f,σ,δ-code C over A whose generating polynomial is both a right and left divisor of f. This gives Theorem 2 a practicality advantage over ([8], Corollary 1). Furthermore, for a monic principal σ-code (resp. a monic principal σ-constacyclic code) over A, the control matrix given in Theorem 2 takes a better shape; see Corollary 6 (resp. Corollary 7).(ii)
For a finite commutative ringA and an automorphism σ of A, the following is performed:(1)
In Section5, we characterize the monic principal σ-codes over A whose dual codes are also monic principal σ-codes, strengthening and extending ([3], Theorem 1); see Theorem 3 and the paragraph that precedes it. Consequently, we give in Corollary 8 a generator matrix of the dual of a monic principal σ-constacyclic code over A, and we further introduce, in Corollary 9, a characterization of self-dual monic principal σ-codes over A in such a way that generalizes and strengthens ([2], Corollary 4).(2)
In Section6, we begin by introducing the notion of a parity-check matrix of a free f,σ,δ-code over a general ring with an endomorphism σ. We then go back to the assumptions on the ring A being finite and commutative and σ an automorphism of A, where we construct a parity-check matrix of a monic principal f,σ,δ-code C over A showing also how to extract such a matrix from a control matrix of C; see Theorem 4. Furthermore, for a monic principal σ-code (resp. a monic principal σ-constacyclic code) over A, the parity-check matrix given in Theorem 4 takes a better shape; see Corollary 10 (resp. Corollary 11). On the other hand, with the crucial assumption that σ is an automorphism of A, we show in Corollary 12 that the parity-check matrix given in Corollary 11 can be obtained without the assumption that the monic principal σ-constacyclic code is generated by some monic g∈Aσ that is also a left divisor of Xn−a.(iii)
Throughout the article, a good number of highlighting examples is given. An earlier preprint of this article is in reference [14]. Some results from this article were used in [15] to construct novel matrix-product codes arising from σ,δ-codes. Other applications of skew codes over rings can be found in ([12], Chapter 6) and ([13], Chapter 11) and the references therein.


## 2. Preliminaries

LetA be a ring with identity, σ a ring endomorphism of A that maps the identity to itself, δ a σ-derivation of A, and UA the multiplicative group of units of A. Fix a monic skew-polynomial fX=∑i=0naiXi∈Aσ,δ of degree n. In order to define the notion of a skew f,σ,δ-code, we begin by using f to endow An with a structure of a left Aσ,δ-module. Let Cf be the usual companion matrix of f; that is,(2)Cf=0100…000010…00⋮⋮⋮⋮⋮⋮⋮0000…01−a0−a1………−an−2−an−1.The mapTf:An⟶An defined by(3)Tfx0,…,xn−1=σx0,…,σxn−1Cf+δx0,…,δxn−1,is a σ,δ-pseudo-linear transformation (associated tof); that is, considering An as a left A-module, we have Tfax=σaTfx+δax for all a∈A and x∈An. It can also be easily checked that Tf is a group endomorphism of An (see [8] for more details and examples on this notion). For a skew-polynomial PX=∑i=0n−1biXi∈Aσ,δ, the map PTf=∑i=0n−1biTfi is obviously a group endomorphism of An as well. Now, the map PX,c0,…,cn−1↦PTfc0,…,cn−1 defines a left action of Aσ,δ on An, which in turn endows An with a left Aσ,δ-module structure as desired.Letfl denote the principal left ideal of Aσ,δ generated by f. With An and Aσ,δ/fl as left Aσ,δ-modules, the map ϕf:An⟶Aσ,δ/fl defined by d0,…,dn−1↦∑i=0n−1diXi+fl is a left Aσ,δ-module isomorphism. The coset ∑i=0n−1diXi+fl is called the polynomial representation of d0,…,dn−1 in Aσ,δ/fl. On the other hand, we know that for each tX∈Aσ,δ, there exists a unique pX=∑i=0n−1diXi∈Aσ,δ of degree at most n−1 such that tX+fl=pX+fl. The n-tuple d0,…,dn−1∈An is called the coordinates of tX+fl (with respect to the basis ℬ=1+fl,X+fl,…,Xn−1+fl). Note that d0,…,dn−1=ϕf−1tX+fl.A skewf,σ,δ-code (or an f,σ,δ-code for short) of length n over A is a linear code C⊆An such that x0,…,xn−1∈C implies that Tfx0,…,xn−1∈C (see [8]). With the above notation, an f,σ,δ-code of lengthnoverA is a subset C of An consisting of the coordinates of a left Aσ,δ-submodule ℳ of Aσ,δ/fl with respect to ℬ, i.e., C=ϕf−1ℳ for some left Aσ,δ-submodule ℳ of Aσ,δ/fl. Equivalently, C⊆An is an f,σ,δ-code if and only if the set ϕfC of polynomial representations of elements of C is a left Aσ,δ-submodule of Aσ,δ/fl. So, there is a one-to-one correspondence between f,σ,δ-codes over A and left Aσ,δ-submodules of Aσ,δ/fl. If δ=0, an f,σ,δ-code may be called an f,σ-code, or just a σ-code if f is irrelevant to the context. A linear code C⊆An is called a σ,δ-code of length n if there exists a monic skew-polynomial f∈Aσ,δ of degree n such that C is an f,σ,δ-code.A ring over which every two bases of any finitely generated free (right) module have the same (finite) number of elements is said to have (right)Invariant Basis Number (IBN for short). This common number is defined to be the rank of such a module. Examples of such rings include nonzero commutative rings, nonzero finite rings, division rings, and local rings. For more on IBN rings, see ([16], Chapter 1). From now on, whenever we mention the finite rank of a free module, we implicitly assume without mention that the underlying ring has IBN.Asℳ is a left Aσ,δ-submodule of Aσ,δ/fl, C is a left Aσ,δ-submodule of An. Then, note a priori that ℳ and C are left A-modules, and ℳ is free over A of rank r if and only if C is free over A of rank r.We call anf,σ,δ-code C=ϕf−1ℳ over Amonic principal if the left Aσ,δ-submodule ℳ of Aσ,δ/fl is generated by a right divisor g∈Aσ,δ of f whose leading coefficient is a unit u; in which case ℳ=gl/fl=u−1gl/fl, where u−1g is obviously monic, and so we can equivalently assume sometimes that ℳ is generated by a monic right divisor of f. Note that if g,h∈Aσ,δ are such that f=hg with a unit leading coefficient of g, then, the leading coefficient of h is a unit as well and degh=degf−degg. On the other hand, a linear code C⊆An is called a monic principal σ,δ-code of length n if there exists a monic skew-polynomial f∈Aσ,δ of degree n such that C is a monic principal f,σ,δ-code. Rephrased according to our terminology, ([8], Theorem 1) shows that a monic principal f,σ,δ-code generated by a monic skew-polynomial g is free over A of rank equal to degf−degg. It should be noted, however, that not all f,σ,δ-codes are monic principal since not all left Aσ,δ-submodules of Aσ,δ/fl are principal to begin with, and even if a left Aσ,δ-submodule of Aσ,δ/fl happens to be a principal submodule generated by a right divisor g of f, g may have a non-unit leading coefficient (unless A has no zero divisors). Being free codes has played an important factor on our choice of working with monic principal f,σ,δ-codes in this article. As such, we could deal with their related notions of generator matrices (Section 3), control matrices (Section 4), and under certain extra conditions, their free dual codes (Section 5) and parity-check matrices (Section 6).In the special case, whenfX=Xn−a for some a∈UA and ℳ is a left Aσ,δ-submodule (resp. a principal left Aσ,δ-submodule generated by a right divisor of Xn−a whose leading coefficient is a unit) of Aσ,δ/Xn−al, the Xn−a,σ,δ-code C=ϕXn−a−1ℳ is called an Xn−a,σ,δ-constacyclic (resp. a monic principal Xn−a,σ,δ-constacyclic) code. In this paper, we deal with such a code only when δ=0 and thus call it an Xn−a,σ-constacyclic (resp. a monic principal Xn−a,σ-constacyclic) code. A linear code C⊆An is called a σ-constacyclic code (resp. a monic principal σ-constacyclic code) of length n if there exists a∈UA such that C is an Xn−a,σ-code (resp. a monic principal Xn−a,σ-code). A monic principal σ-constacyclic code generated by a right divisor g∈Aσ of Xn−a, for some a∈UA, is denoted by gn,σa.


## 3. Generator Matrix of a Monic Principalf,σ,δ-Code over a Ring

In this section, we assume thatA is a ring with identity, σ is a ring endomorphism of A that maps the identity to itself, and δ is a σ -derivation of A. For an A-free f,σ,δ-code C of rank n−r, define a generator matrix of C as a matrix G∈Mn−r,nA whose rows form an A-basis of C (see [17]) for the classical definition of a generator matrix of a linear code over a field). In set notation, we have C=xG|x∈An−r.LetfX=∑i=0naiXi∈Aσ,δ be monic and C a monic principal f,σ,δ-code generated by a monic gX=∑i=0rgiXi∈Aσ,δ of degree r. Then, by ([8], Theorem 1), C is free over A of rank n−r. Using g and the map Tf introduced in Section 1, Boulagouaz and Leroy [8] gave a way of computing G as in Lemma 1. The main aim of this section is to introduce, in Theorem 1, more practical recursive formulas that compute the entries of G using g, σ, and δ. Corollary 5 deals with the case when δ=0.Lemma 1 (see [8], Theorem 1).
With the assumptions as above, the monic principalf,σ,δ-codeChas a generator matrixG∈Mn−r,nAwhose rows are given by(4)Tfig0,…,gr,0,…,0,for 0≤i≤n−r−1.The following results aim at giving the set-up for producing formulas that compute(5)Tfig0,…,gr,0,…,0,much more easily, which among other things gives an obvious programming advantage to the process of computing G, for instance.To simplify notation, fori≥0 and x0,…,xn−1∈An, we set(6)x0i,…,xn−1i=Tfix0,…,xn−1.Lemma 2.
Forx0,…,xn−1∈An and i∈ℕ, we have(a)
x0i=δx0i−1−σxn−1i−1a0;(b)
xji=δxji−1+σxj−1i−1−σxn−1i−1aj, for1≤j≤n−1.Proof.
By definition,(7)Tfx0,x1,…,xn−1=σx0,σx1,…,σxn−1Cf+δx0,δx1,…,δxn−1=δx0−σxn−1a0,δx1+σx0−σxn−1a1,…,δxn−1+σxn−2−σxn−1an−1.As x0i,x1i,…,xn−1i=Tfix0,…,xn−1=Tfx0i−1,x1i−1,…,xn−1i−1, we have(8)x0i,x1i,…,xn−1i=δx0i−1−σxn−1i−1a0,δx1i−1+σx0i−1−σxn−1i−1a1,…,δxn−1i−1+σxn−2i−1−σxn−1i−1an−1.Corollary 1.
Forx0,…,xn−1∈Anwithxn−1=0, we have(a)
x01=δx0;(b)
xj1=δxj+σxj−1for1≤j≤n−2;(c)
xn−11=σxn−2.Proof.
This follows directly from Lemma2 and properties of σ and δ.Corollary 2.
Forx0,…,xn−1∈Anwithxs+1=…=xn−1=0for some0≤s≤n−2, we have(a)
xs+ii=σixsfor1≤i<n−s−1;(b)
xs+ji=0for1≤i<j≤n−s−1.Proof.
We proceed by (finite) induction oni. For i=1, it follows from Corollary 1 that(9)xs+11=δxs+1+σxs=δ0+σxs=σxs.
For1=i<j≤n−s−1, we have s+1≤s+j−1≤n−2 and (by Corollary 1)(10)xs+j1=δxs+j+σxs+j−1=δ0+σ0=0.
Assume now, for1<i<n−s−1, that xs+i−1i−1=σi−1xs and, for i−1<t≤n−s−1, that xs+ti−1=0. Then, it follows from Lemma 2 that(11)xs+ii=δxs+ii−1+σxs+i−1i−1−σxn−1i−1as+i=δ0+σσi−1xs−σ0as+i=σixs.
We also have (by Lemma2), for 1<i<j≤n−s−1,(12)xs+ji=δxs+ji−1+σxs+j−1i−1−σxn−1i−1as+j=δ0+σxs+j−1i−1−σ0as+j=σxs+j−1i−1.
Asi−1<j−1, xs+j−1i−1=0 by assumption. Thus, xs+ji=0 as claimed.Corollary 3.
Letx0,…,xn−1∈Anandδ=0.(a)
Ifxn−1=0, then,x01=0andxj1=σxj−1for1≤j≤n−1.(b)
Ifxs+1=…=xn−1=0for some0≤s≤n−2, then for any1≤i≤n−s−1,(i)
xji=0for0≤j≤i−1, and(ii)
xji=σxj−1i−1for0≤i−1<j≤n−1.Proof.
(a)
A direct application of Corollary1 yields the claim.(b)
We proceed by (finite) induction oni. Let i=1. If 0≤j≤i−1, then, j=0. So, x0i=x01=0 by part (1) above. From part (1) again, for 0=i−1<j≤n−1, xji=xj1=σxj−1=σxj−10=σxj−1i−1 as desired. Assume now that the result holds for all 1≤i<n−s−1. Set yj=xji for each 0≤j≤n−1, and note that yjt=xjit=xji+t for all t≥1. By the inductive assumption, we see that(13)yn−1=xn−1i=σxn−2i−1=σ2xn−3i−2=⋯=σixn−1−i0=σixn−1−i.
Asi<n−s−1, n−1−i>s. So, xn−1−i=0 and, thus, yn−1=0. It now follows from part (1) applied to y0,…,yn−1 that x0i+1=y01=0, and for 1≤j≤n−1, xji+1=yj1=σyj−1=σxj−1i. Note, in particular, that for 1≤j≤i+1, 0≤j−1≤i. So, xj−1i=0 by the inductive assumption, and therefore, xji+1=σ0=0 in this case.Corollary 4.
Letx0,…,xn−1∈An,δ=0, anda1=⋯=an−1=0. Then,(a)
Fori∈ℕ, we have(1)
x0i=−σxn−1i−1a0;(2)
xji=σxj−1i−1for1≤j≤n−1.(b)
If, further,x0=x1=…xs=0for some0≤s≤n−2, then, we have(i)
(1)x01=−σxn−1a0(2)xj1=0for1≤j≤s+1(3)xj1=σxj−1 for s+2≤j≤n−1(ii)
For2≤i≤n−s−1, we have(1)x0i=−σxn−1i−1a0(2)xji=σxj−1i−1 for 1≤j≤i−1(3)Ifs≥1, thenxji=0fori≤j≤i+s−1(4)xji=σxj−1i−1 for i+s≤j≤n−1.Proof.
(a)
This follows directly from Lemma2 with δ=0 and a1=⋯=an−1=0.(b)
By part (a), we have(i)
x01=−σxn−10a0=−σxn−1a0(2)
For1≤j≤s+1, xj1=σxj−10=σxj−1=σ0=0(3)
Fors+2≤j≤n−1, xj1=σxj−10=σxj−1.(ii)
Items 1, 2, and 4 are immediate from part (a). As for item 3, assume thats≥1. We use (finite) induction on i. For i=2 and 2≤j≤s+1, we have 1≤j−1≤s and it thus follows from part (a) and part (b-i-2) that xj2=σxj−11=σ0=0. Suppose now that xji=0 for 2≤i≤n−s−2 and i≤j≤i+s−1. Then, for i+1≤j≤i+s, we have i≤j−1≤i+s−1. So, it follows from part (a) and the inductive step that xji+1=σxj−1i=σ0=0.
Now comes the main result of this section, which gives precise recursive formulas for the entries of a generator matrix ofC enhancing ([8], Theorem 1).Theorem 1.
Keep the assumptions mentioned at the beginning of this section. Then, a generator matrixG∈Mn−r,nAofCis(14)g0…gr00…0g01…gr1σgr0…0g02…gr2gr+12σ2gr…0⋮⋮⋮⋮⋮⋮⋮g0n−r−1…grn−r−1gr+1n−r−1gr+2n−r−1…σn−r−1gr,where (1)
gj=0 for r+1≤j≤n−1,(2)
g0i=δg0i−1 for 1≤i≤n−r−1(3)
gji=δgji−1+σgj−1i−1 for 1≤i≤n−r−1and1≤j≤n−1.Proof.
Using Lemma1 and applying Corollaries 1 and 2 with s=r, x0,…,xn−1=g0,…,gn−1, and gr+1=⋯=gn=0 yield the claim of the theorem.IfC of Theorem 1 is a monic principal σ-code (i.e., δ=0), then, a generator matrix of C takes a more beautiful form as the following result shows, the proof of which is just a direct application of Theorem 1 in this special case.Corollary 5.
Keep all the assumptions of Theorem1 with δ=0. Then, a generator matrix G∈Mn−r,nA of C is(15)g0⋯gr00⋯00σg0⋯σgr0…0⋮⋱⋱⋮00⋯0σn−r−1g0⋯σn−r−1gr.Example 1.
LetR be a ring with identity and A the ring ab0a|a,b∈R. Letting σ:ab0a↦a00a and δ:ab0a↦0b00, it can be checked that σ is a ring endomorphism of A that maps the identity to itself and δ is a σ-derivation of A. Let C a monic principal σ,δ-code of length 4 generated by gX=X−α∈Aσ,δ with α=1101. Noting that g0=−α, g1=1, g2=g3=0, we get from Theorem 1 that a generator matrix of C is(16)G=g0g100g01g11σg10g02g12g22σ2g1=g0100δg0δ1+σg010δg01δ1+σg011101=g0100δg0δg1+σg010δ2g0δ21+δσ+σδg0δσ+σδ1+σ2g01=−α100δ−ασ−α10δ2−α0σ2−α1=−α1001−α−1101−α0−11.
On the other hand, ifδ=0, then, it follows from Corollary 5 that a generator matrix of C is(17)G=g0g1000σg0σg1000σ2g0σ2g1=−α1000−11000−11.Example 2.
LetA=F3×F3, σx,y=y,x, and fX=X6+1∈Aσ. Denoting a,a∈A by a, we can see that fX=X2+1X4+2X2+1=X4+2X2+1X2+1. The σ-code generated by gX=X4+2X2+1 is a monic principal σ-constacyclic (or negacyclic if one wishes), which is a self-orthogonal 6,4,2 code over A with generator matrix 102010010201. Using the obvious Gray map on Magma ([18]), this code yields a ternary 12,4,3 code whose dual is a 12,8,2 code, which is quasioptimal (see [19]).


## 4. Control Matrix of a Monic Principalf,σ,δ-Code over a Ring

In this section, we assume thatA is a ring with identity, σ is a ring endomorphism of A that maps the identity to itself, and δ is a σ-derivation of A. The results of Section 3 are utilized here to give control matrices (defined below) of a monic principal f,σ,δ-code, a monic principal σ-code, and a monic principal σ-constacyclic code.ForH∈Mn,tA with t≤n, denote by AnnlH the left A-submodule of An:(18)AnnlH:=x∈An|xH=0.IfC is an f,σ,δ-code of length n over A, a matrix H∈Mn,tA, with t≤n, is called a control matrix of C if C=AnnlH. Consequently, for an A-free code C, if G is a generator matrix of C and H is a control matrix of C, then, GH=0.For a monic principalf,σ,δ-code C over A that is generated by some monic g∈Aσ,δ which is both a right and left divisor of f, Boulagouaz and Leroy [8] gave a way of computing a control matrix of C, as in Lemma 3, using Tf and h, where h∈Aσ,δ is such that gh=f. Theorem 2 gives precise and more practical recursive formulas that compute a control matrix of C using f, h, σ and δ. Corollary 6 deals with the special case when δ=0, while Corollary 7 handles the more special case when C is monic principal σ-constacyclic.Lemma 3 (see [8], Corollary 1).
LetCbe a monic principalf,σ,δ-code of lengthngenerated by some monicg∈Aσ,δof degreen−kwhich is also a left divisor off, withf=ghfor somehX=∑i=0khiXi∈Aσ,δ. Then, a control matrix ofCis the matrixH∈Mn,nAwhose rows areTfih0,…,hk,0,…,0for0≤i≤n−1.Remark 1.
Lemma3 is still valid if we assume that the leading coefficient of g is a unit in A.
With the assumptions of Lemma3, the following theorem gives explicit and more practical recursive formulas to compute a control matrix.Theorem 2.
Keep the assumptions of Lemma3 with fX=∑i=0naiXi. Then, a control matrix H∈Mn,nAofC is given by(19)h0…hk00…0h01…hk1σhk0…0h02…hk2hk+12σ2hk…0⋮⋮⋮⋮⋮⋮⋮h0n−k−1…hkn−k−1hk+1n−k−1hk+2n−k−1…σn−k−1hkh0n−k…hkn−khk+1n−khk+2n−k…hn−1n−k⋮⋮⋮⋮⋮⋮⋮h0n−1…hkn−1hk+1n−1hk+2n−1…hn−1n−1,
where,(1)
hj=0 for k+1≤j≤n−1,(2)
for2≤n−k, 1≤i≤n−k−1, and 1≤j≤n−1,(i)
h0i=δh0i−1,(ii)
hji=δhji−1+σhj−1i−1,(3)
forn−k≤i≤n−1 and 1≤j≤n−1(i)
h0i=δh0i−1−σhn−1i−1a0, and(ii)
hji=δhji−1+σhj−1i−1−σhn−1i−1aj.Proof.
By Lemma3, a control matrix of C is the matrix H∈Mn,nA whose rows are(20)Tfih0,…,hk,0,…,0=h0i,…,hn−1i,for0≤i≤n−1.
Now applying Lemma2 and Corollaries 1 and 2 with s=k and h0,…,hn−1 in place of x0,…,xn−1 yields the desired conclusion.Remark 2.
In Theorem2, case (1) deals with the first row of H, case (2) deals with the rows (beyond the first row) which end with consecutive zeros, and case (3) deals with the remaining rows. It is obvious that in case n−k=1, we disregard case (2) and consider only cases (1) and (3). In such a case, as n=k+1, the last column of H is the k+1st which has hk at the top, and so the upper triangle of zeros does not exist (see the example below).Example 3.
LetR be a ring of characteristic 3 with identity and A=ab0a|a,b∈R. Take σ:ab0a↦ab0a and δ:ab0a↦0b00. Let fX=X3+2X∈Aσ,δ where 2 obviously denotes 21001. Consider gX=X+2β∈Aσ,δ and hX=X2+βX+α∈Aσ,δ with α=0100 and β=1101. A simple verification shows that f=gh=hg. Let C be the monic principal f,σ,δ-code generated by g. Noting that h0=α, h1=β, and h2=1, it follows from Theorem 2 (cases (1) and (3); see the above remark) that a control matrix of C is(21)H=h0h1h2h01h11h21h02h12h22=αβ1α2α+1βαβ1.
To double-check thatH is a correct control matrix, it follows from Theorem 1 that a generator matrix of C is G=2β102α2β1. Making use of the characteristic of R and properties of σ, δ, α, and β, it is straightforward to check that GH=0.Corollary 6.
Keep the assumptions of Theorem2 with δ=0. Then, a control matrix H∈Mn,nA of C is given by(22)h0h1h2…hk000…00σh0σh2…σhk−1σhk00…000σ2h0…σ2hk−2σ2hk−1σ2hk0…0…………………………00……0σn−k−1h0………σn−k−1hkh0n−kh1n−k……hkn−khk+1n−khk+2n−k……hn−1n−k…………………………h0n−1h1n−1……hkn−1hk+1n−1hk+2n−1……hn−1n−1,where the number of initial consecutive zeros in the ith row is precisely i−1 for i=2,…,n−k, and(1)
hj=0 for k+1≤j≤n−1,(2)
for2≤n−k,1≤i≤n−k−1, and 1≤j≤n−1,(i)
h0i=0,(ii)
hji=σhj−1i−1,(3)
forn−k≤i≤n−1 and 1≤j≤n−1,(i)
h0i=−σhn−1i−1a0, and(ii)
hji=σhj−1i−1−σhn−1i−1aj.Proof.
Use Theorem2 and Corollary 3.Corollary 7.
Keep the assumptions of Corollary6. Let C be a monic principal σ-constacyclic code C=gn,σa for some a∈UA such that g is also a left divisor of Xn−a with Xn−a=gXhX for some hX=∑i=0khiXi∈Aσ,δ. Then, the entries of a control matrix H=Hi,j∈Mn,nAofC are as follows:(a)
Ifn−k=1, then,(23)H1,j=hj−1;if1≤j≤n,and,for2≤i≤n,, then,(24)Hi,j=−σi−1hn−i+ja;if1≤j≤i−1,σi−1hj−i;ifi≤j≤n.(b)
Ifn−k≥2, then,(i)
fori=1,(25)H1,j=hj−1;if1≤j≤k+1,0;ifk+2≤j≤n.(ii)
for2≤i≤n−k,(26)Hi,j=0;if1≤j≤i−1,σi−1hj−i;ifi≤j≤i+k,0;ifi+k+1≤j≤n.(iii)
forn−k+1≤i≤n,(27)Hi,j=−σi−1hn−i+ja;if1≤j≤i−n−k,0;ifi−n−k+1≤j≤i−1,σi−1hj−i;ifi≤j≤n.Proof.
Apply Corollary4 and Corollary 6.


## 5. The Dual of a Monic Principalσ-Code over a Finite Commutative Ring

In this section, we assume thatA is a finite commutative ring with identity and σ is an automorphism of A. We give in Theorem 3 a characterization of monic principal σ-codes over A whose duals are also monic principal σ-codes, strengthening and extending ([3], Theorem 1). Furthermore, Corollary 8 utilizes Theorem 3 and Corollary 5 to give a generator matrix of the dual of a monic principal σ-constacyclic code. Finally, Corollary 9 characterizes self-dual monic principal σ-codes over A in such a way that generalizes and strengthens ([2], Corollary 4).For a linear codeC⊆An, the set y∈An|x,y=0for allx∈C of elements of An orthogonal to C with respect to the Euclidean inner product on An is called the dual of C and is denoted by C⊥. It can be checked that C⊥ is a left A-submodule of An and so is a linear code. It is noted that if C is free with a generator matrix G and a control matrix H, then, it follows from the equality GH=0 (see Section 4) that the columns of H are elements of the dual code C⊥.For a skew-polynomialhX=∑i=0shiXi∈Aσ, define the following skew-polynomials:(28)σnhX=∑i=0sσnhiXiforn∈ℕandh∗X=∑i=0sσihs−iXi.Consider the ring of Laurent skew-polynomials:(29)AX,X−1;σ=∑i=−mnaiXi|m,n∈ℕ∪0,ai∈A,where addition is given by the usual rule and multiplication is given by the rule:(30)aiXibjXj=aiσibjXi+jfori,j∈ℤ,and then extending associatively and distributively to all elements of AX,X−1;σ. It is obvious that Aσ is a subring of AX,X−1;σ. It is worth noting that X−1a=σ−1aX−1 and aX−1=X−1σa for all a∈A.The following result and its proof are similar, in part, to their counterparts over finite fields appearing in the literature (see for instance [3], Lemma 1]).Lemma 4.
Letψ:AX,X−1;σ⟶AX,X−1;σbe the map defined by(31)∑i=−mnaiXi↦∑i=−mnX−iai,
Also, lethX=∑i=0shiXi∈Aσ be of degree s. Then, the following holds:(i)
ψ is a ring antiautomorphism,(ii)
h∗X=XsψhX,(iii)
for anyn∈ℕ,XnhX=σnhXXn, and(iv)
ifhs is not a zero divisor in A, then, hX is not a zero divisor in Aσ.Proof.
(i)
It is straightforward to show thatψ is bijective and additive. Consider two Laurent skew-polynomials SX=∑i=−m1n1siXi and TX=∑j=−m2n2tjXj. Letting k=maxm1,m2, we may add zero terms if necessary to set SX=∑i=−kn1siXi and TX=∑j=−kn2tjXj. Then,(32)ψSXTX=ψ∑i=−kn1siXi∑j=−kn2tjXj=ψ∑i=−kn1∑j=−kn2siXitjXj=ψ∑j=−kn2∑i=−kn1siσitjXi+j=∑j=−kn2∑i=−kn1X−i+jsiσitj.On the other hand,(33)ψTXψSX=∑j=−kn2X−jtj∑i=−kn1X−isi=∑j=−kn2∑i=−kn1X−jtjX−isi=∑j=−kn2∑i=−kn1X−i+jσitjsi=∑j=−kn2∑i=−kn1X−i+jsiσitj.Thus,ψSXTX=ψTXψSX.(ii)
We see that(34)XsψhX=Xs∑i=0sX−ihi=∑i=0sXs−ihi=∑i=0sσs−ihiXs−i=∑j=0sσjhs−jXj=h∗X.(iii)
XnhX=∑i=0sXnhiXi=∑i=0sσnhiXi+n=σnhXXn.(iv)
LetrX=∑j=0trjXj∈Aσ be such that rt≠0 and hXrX=0 (resp. rXhX=0). Then, σsrths=0 (resp. rtσths=0). Note that since hs is not a zero divisor in A and σt is an automorphism of A, σths is not a zero divisor in A either. It then follows that σsrt=0 (resp. rt=0). Since σs is an automorphism of A, it follows in both cases that rt=0, a contradiction. Thus, rX=0.
Special cases of the following two results (in the context of finite fields) appear in [3].Lemma 5.
LetgX=∑i=0n−kgiXi∈Aσbe of degreen−k,gn−k∈UA,hX=∑i=0khiXi∈Aσof degreek, andb∈UA. Then,Xn−b=gXhXif and only ifXn−a=σnhXgXfora=σkbσk−ngn−kσkgn−k−1.Proof.
Either of the claimed equivalent statements imply thathk∈UA. We first prove the lemma for the case when g is monic. Assume that Xn−b=gXhX. Then, h is monic too. It follows from Lemma 4 (iii) that,(35)σnhXgXhX=σnhXXn−σnhXb=XnhX−σnhXb.
So,Xn−σnhXgXhX=σnhXb. Since degh=degσnhb and h is monic, degXn−σnhXgX=0 regardless of the characteristic of A. So, Xn−σnhXgX=a for some nonzero a∈A, and ahX−σnhXb=0. Since h and σnh are monic, the leading coefficient of ahX−σnhXb is a−σkb. Thus, a=σkb and Xn−σkb=σnhXgX as claimed. Conversely, suppose that Xn−σkb=σnhXgX. Applying the above argument for σnh and σkb instead of g and b, respectively, yields(36)Xn−σn−kσkb=σngXσnhX.
So,σnXn−b=σngXhX and, thus, Xn−b=gXhX as claimed.
We now drop the assumption thatg is monic. Assume that Xn−b=gXhX and let G=gn−k−1g. Then, G∈Aσ is monic, and(37)GXhX=gn−k−1Xn−gn−k−1b=Xnσ−ngn−k−1−bgn−k−1=Xn−bσ−ngn−kgn−k−1σngn−k−1.
LettingH=hσ−ngn−k∈Aσ, we then have GXHX=Xn−bσ−ngn−kgn−k−1. Since G is monic and bσ−ngn−kgn−k−1∈UA, it follows from the argument in the first paragraph of this proof that(38)Xn−σkbσk−ngn−kσkgn−k−1=Xn−σkbσ−ngn−kgn−k−1=σnHXGX=σnhXgn−kGX=σnhXgX,as claimed.
Conversely, suppose thatXn−a=σnhXgX with a=σkbσk−ngn−kσkgn−k−1. Note that a∈UA since σ is an automorphism of A and gn−k∈UA. Let G=gn−k−1g. Then, G∈Aσ is monic and Xn−a=σnhXgn−kGX. As hgn−k∈Aσ and σk and σn are automorphisms of A (and also additive automorphisms when extended to Aσ ), let c∈UA and H∈Aσ be such that a=σkc and σnhgn−k=σnH. So, Xn−σkc=HXGX. It now follows from the argument in the first paragraph of this proof that Xn−c=GXHX; that is,(39)Xn−σ−ka=GXhXσ−ngn−k=gn−k−1gXhXσ−ngn−k.
So,(40)gn−kXn−σ−ka=gXhXσ−ngn−k,Xnσ−ngn−k−gn−kσ−ka=gXhXσ−ngn−k,Xn−gn−kσ−kaσ−ngn−k−1σ−ngn−k=gXhXσ−ngn−k,Xn−gn−kσ−kaσ−ngn−k−1=gXhXσ−ngn−kσ−ngn−k−1,Xn−gn−kbσ−ngn−kgn−k−1σ−ngn−k−1=gXhX.
Hence,Xn−b=gXhX as claimed.Remark 3.
If we do not want to be so specific about the nature ofa,b, and h as they appear above, we could rephrase Lemma 5 as follows:
A skew-polynomialg∈Aσ, whose leading coefficient is a unit in A, is a left divisor of Xn−b∈Aσ for some b∈UA if and only if g is a right divisor of Xn−a∈Aσ for some a∈UA.Example 4.
Letσ be an automorphism of A, and α∈UA with σα=α. For gX=X−α and hX=X3+αX2+α2X+α3, we have X4−α4=gXhX in Aσ. On the other hand,(41)σ4hXgX=hXgX=X4−σ3α4σ−11σ31−1=X4−α4,as asserted by Lemma 5.Lemma 6.
LethX=∑i=0khiXi∈Aσbe of degreekwithhk,h0∈UA. Ifhis a right divisor ofXn−bfor someb∈UA, then,h∗is a left divisor ofXn−σk−nb−1and a right divisor ofXn−b−1σ−kh0σn−kh0−1.Proof.
Suppose thath is a right divisor of Xn−b for some b∈UA. So, lXhX=Xn−b for some l∈Aσ with degl=n−k (as hk∈UA). We then have from Lemma 4:(42)ψhXψlX=X−n−b,XkψhXψlXXn−k=1−XkbXn−k,h∗XψlXXn−k=1−Xnσk−nb,=σk−nb−1−Xnσk−nb,h∗XψlXXn−kσk−nb−1=σk−nb−1−Xn,h∗X−ψlXXn−kσk−nb−1=Xn−σk−nb−1.
Sincedegl=n−k, −ψlXXn−kσk−nb−1∈Aσ. It is now obvious that h∗ is a left divisor of Xn−σk−nb−1. Now, keeping in mind that degh∗=degh=k and the leading coefficient of h∗ is h0∈UA, it follows from Lemma 5 that h∗ is a right divisor of Xn−a, where(43)a=σn−kσk−nb−1σ−kh0σn−kh0−1=b−1σ−kh0σn−kh0−1,as claimed.Example 5.
Keep the notations of Example4. By Lemma 6, h∗X=α3X3+α2X2+αX+1 is a left divisor of X4−σ−1α−4=X4−α−4. In fact, we have(44)α3X3+α2X2+αX+1α−3X+α−4=X4−α−4.
We also deduce from Lemma6 that h∗ is a right divisor of X4−σ3α3/α4σα3=X4−α−4 too. In fact, α−3X+α−4α3X3+α2X2+αX+1=X4−α−4.
The following is a very important and interesting fact concerning theA-module orthogonal to a free A-module over a finite commutative ring A, where orthogonality is with respect to the Euclidean inner product. This result is a rephrasing of ([20], Proposition 2.9). It should be noted that the authors of [20] assumed that the finite commutative ring is Frobenius. However, going through their proof and the results they utilized, it becomes clear that such an assumption is unnecessary. It is, however, a necessary assumption for the converse of ([20], Proposition 2.9) to hold, which we do not need here (see [20], Remark 2.10) and the few lines following it).Lemma 7.
IfAis a finite commutative ring with identity,Mis a freeA-submodule ofAnof rankk, andM⊥is theA-submodule ofAnorthogonal toMwith respect to the Euclidean inner product onAn, then,M⊥is free of rankn−k.Proof.
LetG∈Mk,nA be a matrix whose rows are the k elements of an A-basis of M. Then, G is a full-row-rank matrix (that is, the rows of G form a linearly independent set). As it is obvious that M⊥=x∈An|Gxt=0, it follows from ([20], Proposition 2.9) that M⊥ is free of rank n−k.In the terminology of this paper, ([3], Theorem 1) characterizes the monic principal σ-codes over a finite field F (with σ an automorphism of F) whose duals are also monic principal σ-codes, extending ([2], Theorem 2). It is claimed in ([3], p. 240) that ([3], Theorem 1) remains valid over finite rings (not even assuming commutativity!) if one assumes that the constant term of g is a unit. Yet, when looking at the proof of ([3], Theorem 1), we see that a crucial underlying assumption is that the dual of a linear code over a finite field is free (as both are vector spaces) and the sum of the dimensions of the two codes is equal to their length. However, the freeness assumption on the dual does not necessarily hold over rings in general even if the original linear code is free, let alone talking about the sum of the dimensions. So, the same proof of ([3], Theorem 1) cannot be adopted for finite rings and, thus, we can not see at the moment how the aforementioned claim can be verified. To the best of the authors’ knowledge, however, it was not until the appearance of ([20], Proposition 2.9) (Lemma 7) three years after the study by Boucher and Ulmer [3] that we were able to extend ([3], Theorem 1) to finite commutative rings (Theorem 3).Theorem 3.
LetAbe a finite commutative ring with identity,σa ring automorphism ofA, andCa monic principalσ-code of lengthngenerated by some monicgX=∑i=0n−kgiXi∈Aσwithg0∈UA.(i)
If the dualC⊥ofCis a monic principalσ-code generated by somehX=∑i=0khiXi∈Aσwithh0,hk∈UA, then,Cis monic principalσ-constacyclic withC=gn,σσkg0σ2khk.(ii)
If for somea∈UA,C=gn,σais monic principalσ-constacyclic, then, the dualC⊥ofCis the monic principalσ-constacyclic codeC⊥=h∗n,σc, wherehX=∑i=0khiXi∈Aσis such thatXn−σ−ka=gXhXwithh0∈UA, andc=σ−ka−1σ−kh0σn−kh0−1.Proof.
(i)
LetC⊥ be a monic principal σ-code generated by some hX=∑i=0khiXi∈Aσ with hk,h0∈UA. Since h0−1h∈Aσ also generates C⊥, we assume that h0=1, set h⊥X=∑i=0kσk−ihk−iXi, and note that h⊥ is monic. We claim that gXh⊥X=Xn−g0σkhk. Suppose that gXh⊥X=∑i=0nciXi. Notice that cn=1 and c0=g0σkhk. To settle the claim, it remains to show that cl=0 for l∈1,…,n−1. Since XigX0≤i≤k−1 and XjhX0≤j≤n−k−1 are A-generators of C and C⊥, respectively, it follows that(45)Xi0gX,Xi1hX=0.For anyi0∈0,…,k−1 and i1∈0,…,n−k−1. So, for every such i0 and i1, we have(46)0=Xi0gX,Xi1hX=∑i=0n−kσi0gixi+i0,∑i=0kσi1hiXi+i1=∑i=0n−kσi0gixi+i0,∑i=i1−i0k+i1−i0σi1hi−i1+i0Xi+i0=∑i=max0,i1−i0minn−k,k+i1−i0σi0giσi1hi−i1+i0=σi0∑i=max0,i1−i0minn−k,k+i1−i0giσi1−i0hi−i1+i0.Sinceσi0 is an automorphism of A, ∑i=max0,i1−i0minn−k,k+i1−i0giσi1−i0hi−i1+i0=0. Let l=k+i1−i0. Then, l∈1,…,n−1 and(47)σihl−i=σiσl−i−khk−l+i=σl−khk−l+i=σi−i0hi−i1+i0.So,(48)0=∑i=max0,i1−i0minn−k,k+i1−i0giσi1−i0hi−i1+i0=∑i=max0,l−kminn−k,lgiσl−khk−l+i=∑i=max0,l−kminn−k,lgiσihl−i=cl,as desired. It now follows from Lemma5 that Xn−σkg0σ2khk=σnh⊥XgX, and hence, C=gn,σσkg0σ2khk is σ-constacyclic.(ii)
Asg is a right divisor of Xn−a whose leading coefficient is a unit, it follows from Lemma 5 that there exists some hX=∑i=0khiXi∈Aσ such that Xn−σ−ka=gXhX. Since g0h0=σ−ka and A is commutative with σ−ka∈UA, h0∈UA. It then follows from Lemma 6 that h∗ is a right divisor of Xn−c with c=σ−ka−1σ−kh0σn−kh0−1. Let C∗=h∗n,σc be the monic principal σ-constacyclic code generated by h∗. We show that C∗=C⊥. As C is a monic principal σ-code generated by g, which is of degree n−k, C is A-free of rank k ([8], Theorem 1). Since A is a finite commutative ring, it follows from Lemma 7 that C⊥ is A-free of rank n−k. On the other hand, as C∗ is a monic principal σ-code generated by h∗, which is of degree k, C∗ is A-free of rank n−k too. So, C∗=C⊥<∞. It, thus, suffices to show that C∗⊆C⊥. Since XigX0≤i≤k−1 and Xjh∗X0≤j≤n−k−1 are A-generators of C and C∗, respectively, it suffices to show that <XigX,Xjh∗X>=0 for each such i and j. An argument like that in part (i) above will do. Hence, C⊥=h∗n,σc.Remark 4.
If we do not want to be so detailed in Theorem3, we would rephrase it as follows (with some obvious additions):
LetA be a finite commutative ring with identity, σ a ring automorphism of A, and C a monic principal σ-code of length n generated by some monic gX=∑i=0n−kgiXi∈Aσ with g0∈UA. Then, the following are equivalent (assuming in each case that the constant term of the generating skew-polynomial is a unit in A):(i)
C⊥ is a monic principal σ-code.(ii)
C⊥ is a monic principal σ-constacyclic code.(iii)
C is a monic principal σ-constacyclic code.
Note that “i⟶iii” is part i of Theorem 3, “iii⟶ii” is part ii of Theorem 3, and “ii⟶i” is trivial.Example 6.
Keep the notations of Examples4 and 5. As X3+αX2+α2X+α3X−α=X4−α4, let C be the monic principal σ-constacyclic code C=X−α4,σα4. It then follows from Theorem 3 that C⊥=α3X3+α2X2+αX+14,σα−4.Remark 5.
Note that in part (ii) of Theorem3, if aσ−ka=σ−kh0σn−kh0−1, then, C⊥ is the monic principal σ-constacyclic code C⊥=h∗n,σa. That is, both C and C⊥ are generated by right divisors of the same polynomial Xn−a.
If aσ-code C is monic principal σ-constacyclic over a finite commutative ring with identity (where σ is an automorphism of the ring), Theorem 3 asserts that the dual code C⊥ is monic principal σ-constacyclic as well. The following theorem gives a generator matrix of the dual code in such a case.Corollary 8.
LetAbe a finite commutative ring with identity,σa ring automorphism ofA,a∈UA, andC=gn,σaa monic principalσ-constacyclic code generated by some monicgX=∑i=0n−kgiXi∈Aσwithg0∈UA. LethX=∑i=0khiXi∈Aσbe such thatgXhX=Xn−σ−ka, as ensured by Theorem 3. Then, a generator matrix H∈Mn−k,nAofC⊥is(49)hkσhk−1…σkh00…00σhkσ2hk−1…σk+1h00…⋮⋮⋮⋮0……0σn−k−1hk…σn−1h0.Proof.
By Theorem3, the dual code C⊥ is a monic principal σ-constacyclic code generated by h∗. Now, applying Corollary 5 yields the desired conclusion.Example 7.
(a)
Keep the notations of Example5. It follows from Corollary 8 that a generator matrix of C⊥ is H=h3σh2σ2h1σ3h0=1αα2α3.(b)
LetA=ab0a|a,b∈ℤ6 and σ:ab0a↦a−b0a. Let α=1101∈UA, gX=X2+α∈Aσ, and hX=X2−α∈Aσ. We then get(50)hXgX=gXhX=X4−α2.
LettingC=g4,σα2, it follows from Theorem 3 that C⊥=h∗4,σα−2 and from Corollary 8 that C⊥ has the following generator matrix:(51)H=h2σh1σ2h000σh2σ2h1σ3h0=1σ0σ2−α00σ1σ20σ3−α=10−α0010−α.
Due to Theorem3, the following result gives a characterization of self-dual σ-codes over finite commutative rings in such a way that generalizes ([2], Corollary 4) and further strengthens it.Corollary 9.
Keep the assumptions of Theorem3 with n=2k. Then, the following statements are equivalent:(i)
Cis a self-dualσ-code.(ii)
Cis a monic principalσ-constacyclic code withC=gn,σa,a∈UA, andσkh0−1h∗=g, wheregXhX=Xn−σ−ka.(iii)
For anyl∈0,…,k,∑i=0lσk−lgigi+k−l=0.Proof.
i⇔ii: assume that C=C⊥. It follows from Theorem 3 and its proof that C⊥ is σ-constacyclic generated by h∗∈Aσ, where hX=∑i=0khiXi is satisfying h0∈UA and gXhX=Xn−σ−ka for some a∈UA. As σkh0−1h∗ also generates C⊥ and both g and σkh0−1h∗ are monic and generate the same code, we must have g=σkh0−1h∗. Conversely, assume that C=gn,σa, for some a∈UA, and σkh0−1h∗=g where gXhX=Xn−σ−ka. Then, by Theorem 3, C⊥ is monic principal and generated by h∗. Since h∗ and σkh0−1h∗=g generate the same code, we conclude that C=C⊥.
i⇔iii: follow the proof of Corollary 4 of [2] verbatim with the use of Theorem 3 and the obvious adjustments.Example 8.
LetA=F3×F3, σx,y=y,x, and denote a,a∈A by a. Taking hX=X2+2X+2∈Aσ, we get h∗X=2X2+2X+1 and σ2h0−1h∗X=22X2+2X+1=X2+X+2. Letting gX=X2+X+2, a simple verification shows that gXhX=X4+1. We then deduce from Corollary 9 (ii) that C=X2+X+24,σ−1 is a self-dual σ-constacyclic code over A, which is negacyclic over A of length 4. Using Magma [18], this yields, after the obvious Gray map, a negacyclic 8,4,3 ternary code over A with the generator matrix in systematic form 10001020010001020010202000010202. So, we get a new construction of the unique self-dual code with these parameters [21], which is classically obtained as a direct sum of two copies of the tetracode [22], (Table XII).Example 9.
ConsiderF4=F2w with w2+w+1=0. Let A=F4×F4, σx,y=x,y2, and fX=X6+1∈Aσ where 1 denotes 1,1. Letting(52)g1X=h1X=X3+1,g2X=1+0,w2x+0,w2x2+x3,h2X=1+0,w2x+0,wx2+x3,g3X=1+0,wx+0,wx2+x3,h3X=1+0,wx+0,w2x2+x3,we see that f=gihi for every i=1,2,3. So, by Corollary 9, the three codes C1=g16,σ−1, C2=g26,σ−1, and C3=g36,σ−1 are self-dual σ-constacyclic codes over A. Generator matrices of C1, C2, and C3 are, respectively, as follows:(53)G1=1,10,00,01,10,00,00,01,10,00,01,10,00,00,01,10,00,01,1,G2=1,10,w20,w21,10,00,00,01,10,w0,w1,10,00,00,01,10,w20,w21,1,G3=1,10,w0,w1,10,00,00,01,10,w20,w21,10,00,00,01,10,w0,w1,1.
Moreover, using the obvious Gray map toF4, we get from C2 a self-dual 12,6,2 code over F4. For this, Magma [18] was used.


## 6. Parity-Check Matrix of a Monic Principalf,σ,δ-Code over a Finite Commutative Ring

LetA be a ring, σ a ring endomorphism of A that maps the identity to itself, and δ a σ-derivation of A. If C is an A-free f,σ,δ-code of length n and rank k, a matrix H∗∈Mn−k,nA is called a parity-check matrix of C if(1)
H∗T is a control matrix of C, and(2)
H∗ is a generator matrix of the dual C⊥.In classical coding theory over finite fields, the dual code of a linear code is also linear, and hence, a parity-check matrix of such a code always exists. However, for a monic principalf,σ,δ-code C over a ring A (despite being A-free), the dual C⊥ may not be A-free, and thus, a parity-check matrix of C may not exist (due to the lack of requirement (2) above). Nonetheless, when A is a finite commutative ring with identity and σ is a ring automorphism of A, nice things happen. With this assumption added to the hypotheses of Theorem 2, Theorem 4 shows that the transpose of the matrix consisting of the last n−k columns of H of Theorem 2 is indeed a parity-check matrix of C. This is a dramatic improvement of Theorem 2 in this important and widely used case.Theorem 4.
LetA be a finite commutative ring with identity, σ a ring automorphism of A, and keep the other notations and assumptions of Theorem 2. Then, a parity-check matrix H∗∈Mn−k,nA of C is given by(54)hkhk1hk2…hkn−k−1hkn−k…hkn−10σhkhk+12…hk+1n−k−1hk+1n−k…hk+1n−100σ2hk…hk+2n−k−1hk+2n−k…hk+2n−1⋮⋮⋮⋮⋮⋮⋮⋮000…σn−k−1hkhn−1n−k…hn−1n−1,wherehjiis as in Theorem 2.Proof.
Note thatH∗ is the transpose of the last n−k columns of H of Theorem 2. The rows H1,…,Hn−k of H∗ are A-linearly independent since H∗ is in echelon form. Let C∗ be the free left A-submodule of An a basis of which is H1,…,Hn−k. Then, C∗ has cardinality equal to An−k. On the other hand, it follows from Lemma 7 that C⊥ is A-free of rank n−k. So, C⊥ has cardinality equal to An−k as well. With H of Theorem 2, we have C=AnnlH⊆AnnlH∗T. By Lemma 7 again, AnnlH∗T is A-free of rank k. Then, AnnlH∗T=Ak=C and so C=AnnlH∗T. Thus, H∗T is a control matrix of C. This, in particular, implies that H1,…,Hn−k∈C⊥. So, C∗⊆C⊥. Since C∗ and C⊥ are of the same finite cardinality, C∗=C⊥. So, H∗ is a generator matrix of C⊥ and thus a parity-check matrix of C.Example 10.
Keep the notation and assumptions of Example3 with A finite and commutative and σ a ring automorphism of A. By Theorem 4, the matrix H∗=1β1 is a parity-check matrix of C. By Theorem 1, a generator matrix of C is G=2β102α2β1. It can be easily checked that GH∗T=0.
Whenδ=0, H∗ takes a nicer form.Corollary 10.
Keep the assumptions of Theorem4 with δ=0. Then, a parity-check matrix H∗∈Mn−k,nA of the monic principal σ-code C is given by(55)hkσhk−1σ2hk−2…σkh0hkk+1hkk+2…hkn−10σhkσ2hk−1…σkh1σk+1h0hk+1k+2…hk+1n−1………………………000……σn−k−1hk……σn−1h0,wherehjiis as in Theorem 2.Proof.
Follows immediately from Theorem4 and Corollary 6.
A special, yet important, case of Corollary10 is when C is a monic principal σ-constacyclic code, in which case H∗ takes a much better form.Corollary 11.
Keep the assumptions of Theorem4withCa monic principalσ-constacyclic code, C=gn,σa for some a∈UA. Then, a parity-check matrix H∗∈Mn−k,nAofC is given by(56)hkσhk−1σ2hk−2…σkh000…00σhkσ2hk−1…σkh1σk+1h00…0………………………000……σn−k−1hk……σn−1h0.Proof.
Forj=1,…,n, let Cj denote the jth column of H in Corollary 7. If n−k=1, then, H∗=Ck+1T, where (by Corollary 7 (a))(57)Ck+1T=CnT=hk,σhk−1,σ2hk−2,…,σkh0.
Suppose now thatn−k≥2. Then, the rows of H∗ are precisely Ck+1T,Ck+2T,…,CnT. We begin by specifying the entries of Ck+1, where we show that(58)Ck+1T=Hi,k+11≤i≤nT=hk,σhk−1,σ2hk−2,…,σkh0,0,…,0,that is, Hi,k+1=σi−1hk−i−1 for 1≤i≤k+1, and Hi,k+1=0 for k+2≤i≤n. By Corollary 7 (b)-(i), H1,k+1=hk. Let 2≤i≤n. We deal with the following three cases:Casej=k+1=n−k: for 2≤i≤n−k=k+1, we have 2≤i≤k+1=j≤i+k, so we are in the second case of Corollary 7 (b)-(ii). Thus, Hi,k+1=σi−1hk−i−1 here. For n−k+1≤i≤n, we have j=k+1=n−k≤i−1≤n−1 and i−n−k+1≤k+1=j. So i−n−k+1≤j≤i−1, and we are in the second case of Corollary 7 (b)-(iii). Thus, Hi,k+1=0 here. This fully verifies the asserted entries of Ck+1 when j=k+1=n−k.Casej=k+1>n−k: for 2≤i≤n−k, we have i≤n−k<j=k+1 and i+k≥2+k>j. So, i≤j≤i+k, and we are in the second case of Corollary 7 (b)-(ii). Thus, Hi,k+1=σi−1hj−1=σi−1hk−i−1 here. Let n−k+1≤i≤n. If i≤j≤n, then, we are in third case of Corollary 7 (b)-(iii). Thus, Hi,k+1=σi−1hj−1=σi−1hk−i−1 here as well. If n−k+1≤j≤i−1, then, i−n−k+1≤n−n−k+1=k+1=j. So i−n−k+1≤j≤i−1, and we are in the second case of Corollary 7 (b)-(iii). Thus, Hi,k+1=0 here. This fully verifies the asserted entries of Ck+1 when j=k+1>n−k.Casej=k+1<n−k: let 2≤i≤n−k. If i≤k+1=j<n−k, then, i+k≤n−k+k=n. So, we have i≤j≤i+k, and we are in the second case of Corollary 7 (b)-(ii). Thus, Hi,k+1=σi−1hk+1−i=σi−1hk−i−1 here. If j=k+1<i≤n−k, then 1≤j≤i−1, and we are in the first case of Corollary 7 (b)-(ii). Thus, Hi,k+1=0 here. For n−k+1≤i≤n, we have j=k+1<n−k<n−k+1≤i. So, j≤i−1. Also, i−n−k+1≤n−n−k+1=k+1=j. So, we have i−n−k+1≤j≤i−1, and we are in the second case of Corollary 7 (b)-(iii). Thus, Hi,k+1=0 here as well. This fully verifies the asserted entries of Ck+1 when j=k+1<n−k.
Now, as forCk+1+t with t=1,…,n−k−1, note that (by Corollary 7) Hi,k+1+t=0 for 1≤i≤t. For t+1≤i≤k+1+t, Corollary 4 (a) yields Hi,k+1+t=σi−1hk+1+t−i. For k+1+t+1≤i≤n, Corollary 4 (a) again yields Hi,k+1+t=0. This completes the proof.
Note that a requirement in the above corollary is thatg be both a right and left divisor of Xn−a (according to Theorem 4). The following corollary deals with the case when g is a right divisor of Xn−a and a left divisor of Xn−σ−ka and g0∈UA (see the assumptions of Corollary 8).Corollary 12.
Keep the assumptions of Corollary8. Then, a parity-check matrix H∗∈Mn−k,nA of C is given by(59)hkσhk−1σ2hk−2…σkh000…00σhkσ2hk−1…σkh1σk+1h00…0………………………000……σn−k−1hk……σn−1h0.Proof.
By Corollary8, H∗ is a generator matrix of C⊥. Furthermore, it is clear that z∈AnnlH∗T if and only if z∈x∈An|x,y=0for ally∈C⊥=C⊥⊥. Since C⊥⊥ and C are both free of the same rank (thanks to Lemma 7) and C⊆C⊥⊥, we conclude that C=AnnlH∗T. Hence, H∗ is a parity-check matrix of C as claimed.


## 7. Conclusion and Future Work

### 7.1. Conclusion

In this article, recursive formulas were provided to compute the entries of generator and control matrices of a monic principalf,σ,δ-code C over a ring A. When A is finite and commutative with an automorphism σ and the generator polynomial of C is both a right and a left divisor of f, a parity-check matrix of C was also given. When further δ=0, a characterization was given for such codes whose dual codes were also monic principal. Particularly, self-dual codes of this kind as well as monic principal skew-constacyclic were discussed.


### 7.2. Future Work

Some of the issues that can be worked on are the following:(i)
Despite the importance of the generator matrices, control matrices, and parity-check matrices in identifying certain monic principal skew codes over rings, improvements on other coding-theoretic parameters are still to be discussed.(ii)
What can be said about monic principal dual codes of monic principal skew codes in caseδ≠0, and what can be said about the non-monic-principal dual codes of monic principal skew codes in both cases δ=0 and δ≠0?(iii)
In caseδ≠0, what can be said about self-dual monic principal σ,δ-codes and constacyclic σ,δ-codes in terms of their characterizations or properties?


## 7.1. Conclusion

In this article, recursive formulas were provided to compute the entries of generator and control matrices of a monic principalf,σ,δ-code C over a ring A. When A is finite and commutative with an automorphism σ and the generator polynomial of C is both a right and a left divisor of f, a parity-check matrix of C was also given. When further δ=0, a characterization was given for such codes whose dual codes were also monic principal. Particularly, self-dual codes of this kind as well as monic principal skew-constacyclic were discussed.


## 7.2. Future Work

Some of the issues that can be worked on are the following:(i)
Despite the importance of the generator matrices, control matrices, and parity-check matrices in identifying certain monic principal skew codes over rings, improvements on other coding-theoretic parameters are still to be discussed.(ii)
What can be said about monic principal dual codes of monic principal skew codes in caseδ≠0, and what can be said about the non-monic-principal dual codes of monic principal skew codes in both cases δ=0 and δ≠0?(iii)
In caseδ≠0, what can be said about self-dual monic principal σ,δ-codes and constacyclic σ,δ-codes in terms of their characterizations or properties?



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