# Skew Constacyclic Codes over Finite Fields and Finite Chain Rings

Authors: Hai Q. Dinh; Bac T. Nguyen; Songsak Sriboonchitta
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

This paper overviews the study of skewΘ-λ-constacyclic codes over finite fields and finite commutative chain rings. The structure of skew Θ-λ-constacyclic codes and their duals are provided. Among other results, we also consider the Euclidean and Hermitian dual codes of skew Θ-cyclic and skew Θ-negacyclic codes over finite chain rings in general and over Fpm+uFpm in particular. Moreover, general decoding procedure for decoding skew BCH codes with designed distance and an algorithm for decoding skew BCH codes are discussed.


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## Body

## 1. Introduction

Reliable communication has been an unavoidable problem for a long time. Before 1948, communication was strictly an engineering discipline. However, there was very little scientific to develop a system to understand it. In 1948, Shannon’s1 landmark paper “A Mathematical Theory of Communication” [1] on the mathematical theory of communication, which showed that good codes exist, gave birth to information theory and coding theory. Coding theory is applicable in many situations that involve a common feature that a sender wants to send a message to a receiver through a noisy-channel. When the receiver has a message, it might contain some errors. Therefore, rather than sending it directly, the sender will encode it and send it to a decoder that estimates the message to give the receiver. Figure 1 describes a communication channel that transmits information from a source to a destination through a system.Figure 1Shannon’s noisy-channel coding theorem ensures that our hopes of getting the correct messages to the users will be fulfilled a certain percentage of the time. Based on the characteristics of the communication channel, it is possible to build the right encoders and decoders so that this percentage, although not 100%, can be made as high as we desire. However, the proof of Shannon’s noisy-channel coding theorem is probabilistic and only guarantees the existence of such good codes. No specific codes were constructed in the proof that provides the desired accuracy for a given channel. The main goal of coding theory is to establish good codes that fulfill the assertions of Shannon’s noisy-channel coding theorem. During the last 50 years, while many good codes have been constructed, but only from 1993, with the introduction of turbo codes2, the rediscoveries of LDPC codes3, and the study of related codes and associated iterative decoding algorithms, researchers started to see codes that approach the expectation of Shannon’s noisy-channel coding theorem in practice.In real life, the noise is unavoidable, so we want toDETECT if there is an error andCORRECT if there is one. In 1950, a colleague of Shannon, Hamming4, developed a ground-breaking idea in his famous paper “Error Detecting and Error Correcting Codes” [2]. The ground-breaking idea in Hamming’s paper describes a single error correcting code.5 A simple extension of this code is also discovered by Hamming in [2]. For more details, we refer the readers to [2].The classes of cyclic and negacyclic codes in particular, and constacyclic codes in general, play a very significant role in the theory of error correcting codes. Allλ-constacyclic codes of length n are classified as ideals fx of F[x]/xn-λ, where f(x) is a divisor of xn-λ. Due to their rich algebraic structure, constacyclic codes can be efficiently encoded using shift registers, which explains their preferred role in engineering.In fact, cyclic codes are the most studied of all codes. Many well-known codes, such as BCH, Kerdock, Golay, Reed-Muller, Preparata, Justesen, and binary Hamming codes, are either cyclic codes or constructed from cyclic codes. Cyclic codes over finite fields were first studied in the late 1950s by Prange [4–7], while negacyclic codes over finite fields were initiated by Berlekamp in the late 1960s [8, 9]. The case when the code length n is divisible by the characteristic p of the field yields the so-called repeated-root codes, which were first studied since 1967 by Berman [10] and then in the 1970s and 1980s by several authors such as Massey et al. [11], Falkner et al. [12], Roth and Seroussi [13], Castagnoli et al. [14], and van Lint [15].In 2007, Boucher et al. initiated [3] the study of skew cyclic codes. They generalized the notion of cyclic codes by using generator polynomials in noncommutative skew polynomial rings. In 2008 and 2011, Boucher and Ulmer [16, 17] continued to study skew Θ-λ-constacyclic codes over Galois rings and codes as modules over skew polynomial rings.In [16], Boucher et al. generalized the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. If finite fields are replaced by Galois rings, then the technical difficulty in studying from finite fields alphabet to Galois rings alphabet is that the skew polynomial rings are not Ore rings. They are neither left nor right Euclidean rings. However, left and right divisor can be defined for some suitable elements. Therefore, in [16], self-dual codes over GR(42) are constructed and used for three applications: self-dual Euclidean codes give self-dual Z4 codes by projection on a trace orthogonal basis, self-dual Hermitian codes build 3-modular lattices, and self-dual Hermitian codes yield self-dual quasi-cyclic codes over Z4 by the cubic construction. For more details, we refer the readers to [16] and the references therein. Boucher and Ulmer also studied the factorization of skew polynomial in skew polynomial rings [18]. These results allowed them to study the skew self-dual cyclic codes with length 2s.The class of finite rings of the formFpm+uFpm has been widely used as alphabets of certain constacyclic codes. For example, the structure of F2+uF2 is interesting; it is lying between F4 and Z4 in the sense that it is additively analogous to F4 and multiplicatively analogous to Z4. It has been studied by a lot of researchers (see, e.g., [19–24]). The classification of codes plays an important role in studying their structures, but, in general, it is very difficult. Only some codes of certain lengths over certain finite fields or finite chain rings are classified. All constacyclic codes of length 2s over the Galois extension rings of F2+uF2 are classified and their detailed structures are also established in [25].In 2012, Jitman et al. [26] introduced the notion of skew Θ-λ-constacyclic (or skew constacyclic) codes over finite chain rings. They studied the structure of skew Θ-λ-constacyclic, the Euclidean, and Hermitian dual codes of skew Θ-cyclic and negacyclic codes over finite chain rings. The goal of this survey is to study skew Θ-λ-constacyclic codes over finite fields and finite chain rings.This paper is arranged as follows. Basic concepts are reviewed in Section2. After presenting preliminary concepts in Section 2, we study skew Θ-negacyclic, cyclic, and Θ-λ-constacyclic codes over finite fields in Section 3. We also introduce some results for Euclidean and Hermitian self-dual codes over finite fields. In Section 4, general decoding procedure for decoding skew BCH codes with designed distance is provided. We also discuss an algorithm for decoding skew BCH codes. Finally, in Section 5, we consider the structure of skew Θ-λ-constacyclic codes over finite chain rings. The Euclidean and Hermitian dual codes over finite chain rings are also exhibited in this section.


## 2. Preliminaries

### 2.1. Finite Fields and Their Automorphisms

In this subsection, we will not give entire properties of finite fields and their automorphisms; rather we will only introduce without proofs some properties of finite fields and their automorphisms that are needed in our consideration later.Definition 1.
LetF be a finite field with multiplicative identity 1. The characteristic of F is the least positive integer p such that p·1=0. Such p always exists for a finite field and it is well known that the characteristic p must be a prime.Theorem 2.
A finite fieldF of characteristic p contains pn elements for some integer n≥1. For every element β of a finite field F with pm elements, we have βpm=β.Definition 3.
An elementα in a finite field Fpm is called aprimitive element (or generator) of Fpm if Fpm={0,α,α2,…,αpm-1}.Example 4.
F
5 has 2 primitive elements, namely, 2 and 3. F4 has 2 primitive elements. In fact, expressing F4 as F2(α)={0,1,α,α+1}, where α2+α+1=0, then α and α+1 are primitive elements of F4.Note that an automorphismφ of a field F is a bijection φ:F→F such that φ(a+b)=φ(a)+φ(b) and φ(ab)=φ(a)φ(b) for all a,b∈F. Suppose that Fpm is a finite field of characteristic p>0, and then the map φp:Fpm→Fpm is defined by φp(x)=xp, the Frobenius automorphism of Fpm. Since Fpm is a field of characteristic p, we have φp(a+b)=(a+b)p=ap+bp=φp(a)+φp(b). From φp(ab)=(ab)p=ap·bp=φp(a)φp(b), we can see that φp is a field homomorphism. Similarly, the map φpt:Fpm→Fpm defined by φpt(x)=xpt is also a field homomorphism. The set of automorphisms of Fpm forms a group under composition which we denote as Aut(Fpm). Next, we give the following theorem characterizing this group.Theorem 5 (see [27, Theorem 3.6.1]).
(i) IfFpm is a finite field, then Aut(Fpm) is a cyclic group of order m and is generated by Frobenius automorphism φp.
(ii) The prime subfield ofFpm is precisely the set of elements in Fpm such that φp(α)=α.
(iii) The subfieldFq of Fqt is precisely the set of elements in Fqt such that φp(α)=α, where q=pm.Theorem 6 (see [27, Theorem 3.6.2]).
IfF1=Fpk⊂F2=Fpn, then Aut(F2/F1) is a cyclic group of order n/k and is generated by φpk.


### 2.2. Codes, Cyclic Codes, Generator, and Parity-Check Matrices

LetFpm be a finite field. A linear (n,k)-code over Fpm is a k-dimensional vector subspace C of the vector space (1)V=Fpmn=a0,…,an-1∣ai∈Fpm.In this paper, all codes are assumed to be linear codes unless otherwise stated. We use polynomial representation of the code C, where we identify codewords (a0,…,an-1)∈C with coefficient tuples of polynomials: (2)c0+c1x+⋯+cn-1xn-1∈Fpmx.Those polynomials can also be seen as elements of a quotient ring Fpm[x]/xn-1, and any code C of length n over Fpm corresponds to a subset of Fpm[x]/xn-1.Example 7.
The polynomialf(x)=a0+a1x+a2x2+⋯+an-1xn-1 of degree at most n-1 over finite field Fpm may be regarded as the word v=a0a1a2⋯an-1 of length n in Fpmn. If n=6, then the polynomial 1+x+x2+x3+x5 may be regarded as the word c=111101. Similarly, the polynomial 1+x3+x4+x5 may be regarded as the word v=100111.Definition 8.
Letc be a word of length n, and the cyclic shift τ(c) the word of length n: (3)τc0,c1,…,cn-1=cn-1,c0,…,cn-2.A code C is said to be cyclic if τ(c)∈C, for all c∈C.Example 9.
LetC={000,111,121,222,012,120,201,021,102,210} be a linear code over Z3. It is easy to see that τ(c)∈C, ∀c∈C. This implies that C is a linear cyclic code over Z3.
LetC1={021,012,000,112,121,100,212,221,200} be a linear code over Z3. Since τ(112)=211∉C1, we can conclude that C1 is not cyclic code.Definition 10.
A codeC is said to be a λ-constacyclic code of length n if it is closed under the λ-constacyclic shift τλ:Fpmn→Fpmn defined by (4)τλc0,c1,…,cn-1=λcn-1,c0,…,cn-2.In particular, when λ=1 or λ=-1, such codes are calledcyclic andnegacyclic codes, respectively.We now give some properties of cyclic code. The following results are well known (cf. [27]).Theorem 11 (see [27, Theorem 4.2.1]).
LetC be a nonzero cyclic code in Fpm[x]/〈xn-1〉. There exists a polynomial g(x)∈C with the following properties:(i)
g(x) is the unique monic polynomial of minimum degree in C, and it is called the generating polynomial for C.(ii)
C=gx.(iii)
The generating polynomialg(x) divides xn-1.(iv)
Ifdeg⁡g(x)=r, then C has dimension n-r and {g(x),xg(x),…,xn-r-1g(x)} is a basis for C.(v)
Every element ofC is uniquely expressible as a product f(x)g(x), where f(x)=0 or deg⁡f(x)<n-r, that is, C=〈g(x)〉={f(x)g(x)∣deg⁡f(x)<n-r}.(vi)
Ifg(x)=g0+g1x+⋯+grxr, then g0≠0 and C has the following generator matrix: (5)G≔g0g1g2⋯gr00⋯00g0g1g2⋯gr0⋯000g0g1g2⋯gr⋯0⋮⋮⋮⋮⋮⋮⋮⋮⋮00⋯0g0g1g2⋯gr⟺gxxgx⋱xn-r-1gx.From this theorem, we can see thatC is a nonzero cyclic code in Fpm[x]/〈xn-1〉 and g(x) is the monic polynomial of minimum degree in C if and only if C=gx, and xn-1 is divisible by g(x).LetC be a cyclic code in Fpm[x]/〈xn-1〉 with generator polynomial g(x), such that deg⁡g(x)=r. Let h(x)=(xn-1)/g(x)=∑i=0n-r‍hixi. Then a parity-check matrix for C is given by (6)H≔hn-r⋯⋯⋯h000⋯00hn-r⋯⋯⋯h00⋯0⋯0hn-r⋯⋯⋯h000⋯⋯⋯⋯⋯⋯⋯⋯⋯000⋯⋯hn-r⋯⋯h0.Example 12.
LetC be a cyclic code of length n=9 over the binary field F2. Put g(x)=x6+x3+1. Then we have h(x)=x3-1. We can see that C has dimension 3 and generating matrix is given by (7)G≔100100100010010010001001001.Hence, a parity-check matrix for C is given by (8)H≔100100000010010000001001000000100100000010010000001001.Definition 13.
(i) The Hamming distancedH(x,y) between two vectors x,y∈Fpmn is defined to be the number of coordinates in which x and y differ.
(ii) The Hamming weightwH(x) of a vector x∈Fpmn is the number of nonzero coordinates in x.
(iii) For a codeC containing at least two words, the minimum distance of a code C, denoted by d(C), is (9)dC=min⁡dx,y,x,y∈C,x≠y.It is easy to see that the definition of distance satisfies nonnegativity, symmetry, and the triangle inequality, so our codeC is living in a metric space.Example 14.
LetC={00000,00111,11111} be a binary code. Then we have (10)dH00000,00111=3;wH00111=3.We can see that (11)d00000,00111=3;d00111,11111=2;d00000,11111=5.This shows that d(C)=2.The following theorem gives a relationship between minimum distanced and the minimum weight of the nonzero codewords of a linear code C.Theorem 15 (see [27, Theorem 1.4.2]).
Ifx,y∈Fpmn, then d(x,y)=wt(x,y). If C is a linear code, the minimum distance d is the same as the minimum weight of the nonzero codewords of C.


### 2.3. The Skew Polynomial RingFpm[x;Θ]

Now letΘ be an automorphism of Fpm. We consider the set Fpm[x;Θ]={a0+a1x+⋯+an-1xn∣ai∈Fpm,n∈N} of formal polynomials where coefficients are written on the left of the variable x. The set Fpm[x;Θ] forms a ring under the usual addition of polynomials and the multiplication is defined by the following basic rule: xa=Θ(a)x. The multiplication is extended to all elements in Fpm[x;Θ] by associativity and distributivity. The ring Fpm[x;Θ] is called askew polynomial ring over Fpm, and each element in Fpm[x;Θ] is called askew polynomial. It is easy to see that the ring Fpm[x;Θ] is noncommutative unless Θ is the identity automorphism on Fpm. If f(x)=a0+a1x+⋯+anxn with an≠0, then we say that f(x) has degree n, denoted by deg⁡(f(x)). The following facts are straightforward for the skew polynomial ring Fpm[x;Θ]:(i)
It has no nonzero zero-divisors,(ii)
the units ofFpm[x;Θ] are the units of Fpm,(iii)
deg⁡(f+g)≤max⁡deg⁡f,deg⁡g,(iv)
deg⁡(f·g)=deg⁡(f)+deg⁡(g).Recall that a left (right) ideal I of a ring R is called aleft (right) principal ideal if there exists an element g∈I such that I=g, where 〈g〉={r·g(g·r):r∈R}. The element g is called agenerator of I and I is said to be generated by g. A ring R is called aleft (right) principal ideal ring if every left (right) ideal of R is principal. The skew polynomial ring Fpm[x;Θ] is left and right Euclidean ring whose left and right ideals are principal. For f(x),g(x)∈Fpm[x;Θ] which are nonzero, there exists unique polynomial h(x),r(x)∈Fpm[x;Θ] such that f(x)=h(x)g(x)+r(x). If r(x)=0, then g(x) is a right divisor of f(x) in Fpm[x;Θ]. The definition of left divisor in Fpm[x;Θ] is similar.The centreZ(Fpm[x;Θ]) of the skew polynomial ring Fpm[x;Θ] is the set of all elements that commute with all other elements of Fpm[x;Θ]. An element f∈Z(Fpmx;Θ) is called acentral element. An automorphism Θ∈Aut(Fpm) is said to fix an element α∈Fpm if Θ(α)=α. We denote FpmΘ⊂Fpm the subfield of elements of Fpm which are fixed by Θ. Then the ring FpmΘ[x] is a commutative subring of Fpm[x;Θ]. A polynomial f∈Fpm[x;Θ] is central element if and only if f is both in FpmΘ and in Fpm[xm], where Fpm[xm]={a0+a1xm+⋯+adxmd,d∈N,ai∈Fpm}. In other words, a polynomial P∈Fpm[x;Θ] is central element (i.e., commutes with all elements of Fpm[x;Θ]) if and only if P=∑i=0m‍ciXiα∈Fpm[x;Θ], where α=Θ is the order of Θ [28, Theorem II.12].


## 2.1. Finite Fields and Their Automorphisms

In this subsection, we will not give entire properties of finite fields and their automorphisms; rather we will only introduce without proofs some properties of finite fields and their automorphisms that are needed in our consideration later.Definition 1.
LetF be a finite field with multiplicative identity 1. The characteristic of F is the least positive integer p such that p·1=0. Such p always exists for a finite field and it is well known that the characteristic p must be a prime.Theorem 2.
A finite fieldF of characteristic p contains pn elements for some integer n≥1. For every element β of a finite field F with pm elements, we have βpm=β.Definition 3.
An elementα in a finite field Fpm is called aprimitive element (or generator) of Fpm if Fpm={0,α,α2,…,αpm-1}.Example 4.
F
5 has 2 primitive elements, namely, 2 and 3. F4 has 2 primitive elements. In fact, expressing F4 as F2(α)={0,1,α,α+1}, where α2+α+1=0, then α and α+1 are primitive elements of F4.Note that an automorphismφ of a field F is a bijection φ:F→F such that φ(a+b)=φ(a)+φ(b) and φ(ab)=φ(a)φ(b) for all a,b∈F. Suppose that Fpm is a finite field of characteristic p>0, and then the map φp:Fpm→Fpm is defined by φp(x)=xp, the Frobenius automorphism of Fpm. Since Fpm is a field of characteristic p, we have φp(a+b)=(a+b)p=ap+bp=φp(a)+φp(b). From φp(ab)=(ab)p=ap·bp=φp(a)φp(b), we can see that φp is a field homomorphism. Similarly, the map φpt:Fpm→Fpm defined by φpt(x)=xpt is also a field homomorphism. The set of automorphisms of Fpm forms a group under composition which we denote as Aut(Fpm). Next, we give the following theorem characterizing this group.Theorem 5 (see [27, Theorem 3.6.1]).
(i) IfFpm is a finite field, then Aut(Fpm) is a cyclic group of order m and is generated by Frobenius automorphism φp.
(ii) The prime subfield ofFpm is precisely the set of elements in Fpm such that φp(α)=α.
(iii) The subfieldFq of Fqt is precisely the set of elements in Fqt such that φp(α)=α, where q=pm.Theorem 6 (see [27, Theorem 3.6.2]).
IfF1=Fpk⊂F2=Fpn, then Aut(F2/F1) is a cyclic group of order n/k and is generated by φpk.


## 2.2. Codes, Cyclic Codes, Generator, and Parity-Check Matrices

LetFpm be a finite field. A linear (n,k)-code over Fpm is a k-dimensional vector subspace C of the vector space (1)V=Fpmn=a0,…,an-1∣ai∈Fpm.In this paper, all codes are assumed to be linear codes unless otherwise stated. We use polynomial representation of the code C, where we identify codewords (a0,…,an-1)∈C with coefficient tuples of polynomials: (2)c0+c1x+⋯+cn-1xn-1∈Fpmx.Those polynomials can also be seen as elements of a quotient ring Fpm[x]/xn-1, and any code C of length n over Fpm corresponds to a subset of Fpm[x]/xn-1.Example 7.
The polynomialf(x)=a0+a1x+a2x2+⋯+an-1xn-1 of degree at most n-1 over finite field Fpm may be regarded as the word v=a0a1a2⋯an-1 of length n in Fpmn. If n=6, then the polynomial 1+x+x2+x3+x5 may be regarded as the word c=111101. Similarly, the polynomial 1+x3+x4+x5 may be regarded as the word v=100111.Definition 8.
Letc be a word of length n, and the cyclic shift τ(c) the word of length n: (3)τc0,c1,…,cn-1=cn-1,c0,…,cn-2.A code C is said to be cyclic if τ(c)∈C, for all c∈C.Example 9.
LetC={000,111,121,222,012,120,201,021,102,210} be a linear code over Z3. It is easy to see that τ(c)∈C, ∀c∈C. This implies that C is a linear cyclic code over Z3.
LetC1={021,012,000,112,121,100,212,221,200} be a linear code over Z3. Since τ(112)=211∉C1, we can conclude that C1 is not cyclic code.Definition 10.
A codeC is said to be a λ-constacyclic code of length n if it is closed under the λ-constacyclic shift τλ:Fpmn→Fpmn defined by (4)τλc0,c1,…,cn-1=λcn-1,c0,…,cn-2.In particular, when λ=1 or λ=-1, such codes are calledcyclic andnegacyclic codes, respectively.We now give some properties of cyclic code. The following results are well known (cf. [27]).Theorem 11 (see [27, Theorem 4.2.1]).
LetC be a nonzero cyclic code in Fpm[x]/〈xn-1〉. There exists a polynomial g(x)∈C with the following properties:(i)
g(x) is the unique monic polynomial of minimum degree in C, and it is called the generating polynomial for C.(ii)
C=gx.(iii)
The generating polynomialg(x) divides xn-1.(iv)
Ifdeg⁡g(x)=r, then C has dimension n-r and {g(x),xg(x),…,xn-r-1g(x)} is a basis for C.(v)
Every element ofC is uniquely expressible as a product f(x)g(x), where f(x)=0 or deg⁡f(x)<n-r, that is, C=〈g(x)〉={f(x)g(x)∣deg⁡f(x)<n-r}.(vi)
Ifg(x)=g0+g1x+⋯+grxr, then g0≠0 and C has the following generator matrix: (5)G≔g0g1g2⋯gr00⋯00g0g1g2⋯gr0⋯000g0g1g2⋯gr⋯0⋮⋮⋮⋮⋮⋮⋮⋮⋮00⋯0g0g1g2⋯gr⟺gxxgx⋱xn-r-1gx.From this theorem, we can see thatC is a nonzero cyclic code in Fpm[x]/〈xn-1〉 and g(x) is the monic polynomial of minimum degree in C if and only if C=gx, and xn-1 is divisible by g(x).LetC be a cyclic code in Fpm[x]/〈xn-1〉 with generator polynomial g(x), such that deg⁡g(x)=r. Let h(x)=(xn-1)/g(x)=∑i=0n-r‍hixi. Then a parity-check matrix for C is given by (6)H≔hn-r⋯⋯⋯h000⋯00hn-r⋯⋯⋯h00⋯0⋯0hn-r⋯⋯⋯h000⋯⋯⋯⋯⋯⋯⋯⋯⋯000⋯⋯hn-r⋯⋯h0.Example 12.
LetC be a cyclic code of length n=9 over the binary field F2. Put g(x)=x6+x3+1. Then we have h(x)=x3-1. We can see that C has dimension 3 and generating matrix is given by (7)G≔100100100010010010001001001.Hence, a parity-check matrix for C is given by (8)H≔100100000010010000001001000000100100000010010000001001.Definition 13.
(i) The Hamming distancedH(x,y) between two vectors x,y∈Fpmn is defined to be the number of coordinates in which x and y differ.
(ii) The Hamming weightwH(x) of a vector x∈Fpmn is the number of nonzero coordinates in x.
(iii) For a codeC containing at least two words, the minimum distance of a code C, denoted by d(C), is (9)dC=min⁡dx,y,x,y∈C,x≠y.It is easy to see that the definition of distance satisfies nonnegativity, symmetry, and the triangle inequality, so our codeC is living in a metric space.Example 14.
LetC={00000,00111,11111} be a binary code. Then we have (10)dH00000,00111=3;wH00111=3.We can see that (11)d00000,00111=3;d00111,11111=2;d00000,11111=5.This shows that d(C)=2.The following theorem gives a relationship between minimum distanced and the minimum weight of the nonzero codewords of a linear code C.Theorem 15 (see [27, Theorem 1.4.2]).
Ifx,y∈Fpmn, then d(x,y)=wt(x,y). If C is a linear code, the minimum distance d is the same as the minimum weight of the nonzero codewords of C.


## 2.3. The Skew Polynomial RingFpm[x;Θ]

Now letΘ be an automorphism of Fpm. We consider the set Fpm[x;Θ]={a0+a1x+⋯+an-1xn∣ai∈Fpm,n∈N} of formal polynomials where coefficients are written on the left of the variable x. The set Fpm[x;Θ] forms a ring under the usual addition of polynomials and the multiplication is defined by the following basic rule: xa=Θ(a)x. The multiplication is extended to all elements in Fpm[x;Θ] by associativity and distributivity. The ring Fpm[x;Θ] is called askew polynomial ring over Fpm, and each element in Fpm[x;Θ] is called askew polynomial. It is easy to see that the ring Fpm[x;Θ] is noncommutative unless Θ is the identity automorphism on Fpm. If f(x)=a0+a1x+⋯+anxn with an≠0, then we say that f(x) has degree n, denoted by deg⁡(f(x)). The following facts are straightforward for the skew polynomial ring Fpm[x;Θ]:(i)
It has no nonzero zero-divisors,(ii)
the units ofFpm[x;Θ] are the units of Fpm,(iii)
deg⁡(f+g)≤max⁡deg⁡f,deg⁡g,(iv)
deg⁡(f·g)=deg⁡(f)+deg⁡(g).Recall that a left (right) ideal I of a ring R is called aleft (right) principal ideal if there exists an element g∈I such that I=g, where 〈g〉={r·g(g·r):r∈R}. The element g is called agenerator of I and I is said to be generated by g. A ring R is called aleft (right) principal ideal ring if every left (right) ideal of R is principal. The skew polynomial ring Fpm[x;Θ] is left and right Euclidean ring whose left and right ideals are principal. For f(x),g(x)∈Fpm[x;Θ] which are nonzero, there exists unique polynomial h(x),r(x)∈Fpm[x;Θ] such that f(x)=h(x)g(x)+r(x). If r(x)=0, then g(x) is a right divisor of f(x) in Fpm[x;Θ]. The definition of left divisor in Fpm[x;Θ] is similar.The centreZ(Fpm[x;Θ]) of the skew polynomial ring Fpm[x;Θ] is the set of all elements that commute with all other elements of Fpm[x;Θ]. An element f∈Z(Fpmx;Θ) is called acentral element. An automorphism Θ∈Aut(Fpm) is said to fix an element α∈Fpm if Θ(α)=α. We denote FpmΘ⊂Fpm the subfield of elements of Fpm which are fixed by Θ. Then the ring FpmΘ[x] is a commutative subring of Fpm[x;Θ]. A polynomial f∈Fpm[x;Θ] is central element if and only if f is both in FpmΘ and in Fpm[xm], where Fpm[xm]={a0+a1xm+⋯+adxmd,d∈N,ai∈Fpm}. In other words, a polynomial P∈Fpm[x;Θ] is central element (i.e., commutes with all elements of Fpm[x;Θ]) if and only if P=∑i=0m‍ciXiα∈Fpm[x;Θ], where α=Θ is the order of Θ [28, Theorem II.12].


## 3. Structure and Duals of Skew Constacyclic Codes over Finite Fields

In this section, we study skewΘ-λ-constacyclic codes over finite fields. We extend the work of Boucher et al. (in 2007) [3] on skew cyclic codes. For more details, we refer the readers to [3] and the references therein. We first introduce the definition of skew Θ-λ-constacyclic codes over finite fields.Definition 16.
Given an automorphismΘ of Fpm and a unit λ in Fpm, a code C is said to beskewΘ-λ-constacyclic of length n if it is closed under the skew Θ-λ-constacyclic shift τΘ,λ:Fpmn→Fpmn defined by (12)τΘ,λc0,c1,…,cn-1=Θλcn-1,Θc0,…,Θcn-2.In particular, whenλ=1 or λ=-1, such codes are calledskewΘ-cyclic andskewΘ-negacyclic codes, respectively. When Θ is the identity automorphism, they become classical constacyclic cyclic, cyclic, and negacyclic codes. A right factor of degree n-k of xn-λ generates [n,k] linear code. While the ring Fpm[x] is a commutative ring, so every ideal in Fpm[x] is two-sided ideal, the skew polynomial ring Fpm[x;Θ] is noncommutative. Therefore, we need to have conditions of Θ and λ to ensure that xn-λ is a two-sided ideal of Fpm[x;Θ]. If n is divisible by the order of Θ and λ is fixed by Θ, then 〈xn-λ〉 is a two-sided ideal of Fpm[x;Θ]. Indeed, for all h(x)=∑i=0t‍hixi in Fpm[x;Θ], we can see that (13)xn-λ∑i=0thixi=Θnh0xn+Θnh1xn+1+⋯+Θnhtxn+t-λ∑i=0thixi.Since n is divisible by the order of Θ, we have Θn(hi)=hi, ∀i={1,…,t}. If λ is fixed by Θ, then we have (xn-λ)(∑i=0t‍hixi)=(∑i=0t‍hixi)(xn-λ), proving that xn-λ is in Z(Fpm[x;Θ]). This implies that 〈xn-λ〉 is a two-sided ideal of Fpm[x;Θ], which makes the quotient ring Fpm[x;Θ]/〈xn-λ〉 well defined. If Θ is not the identity, then Fpm[x;Θ] is in general not a unique factorization ring. In this case, there are typically many more right factors than in the commutative case, producing many Θ-λ-constacyclic codes.Example 17.
Letα be a generator of the multiplicative group of F4; that is, α is a zero of z2+z+1∈F2[z]. Let Θ be the automorphism a↦a2 of F4. We consider the polynomial x6+αx3∈F4[x;Θ]. We have (14)x6+αx3=x4+αxx2=x4+αx3x2+αx+1=x4+αx3+x2x2+αx.This shows that the ring F4[x;Θ] is not a unique factorization ring.Lemma1 in [3] can be extended as follows.Lemma 18 (extending [3, Lemma 1]).
LetΘ be an automorphism of Fpm,n an integer divisible by the order of Θ, and λ a unit in Fpm which is fixed by Θ. The ring Fpm[x;Θ]/〈xn-λ〉 is a principal left ideal ring, in which the left ideals are generated by g(x), where g(x) is a right divisor of xn-λ in Fpm[x;Θ].Consider a codewordc(x)=c0+c1x+⋯+cn-1xn-1. Then(15)xcx=xc0+xc1x+⋯+xcn-1xn-1=Θc0x+Θc1x2+⋯+Θcn-1xn=Θλcn-1+Θc0x+Θc1x2+⋯+Θcn-2xn-1.Thus, xc(x) is corresponding to a Θ-λ-constacyclic shift of c(x), proving that the code C is a skew Θ-λ-constacyclic code if and only if C is a left ideal gx⊆Fpmx;Θ/xn-λ, where g(x) is a right divisor of xn-λ. We summarize this discussion by the following theorem, which is an extension of [3, Theorem 1].Theorem 19 (extending [3, Theorem 1]).
LetΘ be an automorphism of Fpm, n an integer divisible by the order of Θ, and λ a unit in Fpm which is fixed by Θ. Then the code C is a skew Θ-λ-constacyclic code if and only if C is a left ideal 〈g(x)〉⊆Fpm[x;Θ]/〈xn-λ〉, where g(x) is a right divisor of xn-λ.Given a monic right divisor of degreen-k of xn-λ:g(x)=∑i=0n-k-1‍gi(x)+xn-k, then a generator matrix of the Θ-λ-constacyclic code C generated by g(x) is given by (16)G≔g0⋯gn-k-110⋯00Θg0⋯Θgn-k-11⋯00⋯⋯⋯Θ2gn-k-1⋯0⋮⋮⋮⋮⋮⋮⋮0⋯0Θk-1g0⋯Θk-1gn-k-11.Lemma 20 (see [29, Lemma 17]).
LetΘ be an automorphism of Fpm,  n an integer divisible by the order of Θ, and λ a unit in Fpm which is fixed by Θ. Let C be the Θ-λ-constacyclic code generated by a monic right divisor g(x) of xn-λ and h(x)≔(xn-λ)/g(x). If h=h0+h1x+⋯+xn-r, then the following matrix (17)H≔1Θhn-r-1⋯Θn-rh00⋯001Θ2hn-r-1⋯Θn-r+1h0⋯000⋯⋯⋯⋯0⋮⋮⋱⋱⋮⋱⋮00⋯1Θrhn-r-1⋯Θn-1h0is a parity-check matrix for C.SinceΘ(1)=1 for any Θ∈Aut(Fpm), we have Θ(-1)=-1. This shows that -1∈Fpm is fixed by Θ. The following two corollaries are direct consequences of Theorem 19.Corollary 21.
LetΘ be an automorphism of Fpm and n an integer divisible by the order of Θ. Then the code C is a skew Θ-negacyclic code if and only if C is a left ideal gx⊆Fpm[x;Θ]/xn+1, where g(x) is a right divisor of xn+1.Corollary 22 (see [3, Lemma 1]).
LetΘ be an automorphism of Fpm and n an integer divisible by the order of Θ. Then the code C is a skew Θ-cyclic code if and only if C is a left ideal 〈g(x)〉⊆Fpm[x;Θ]/xn-1, where g(x) is a right divisor of xn-1.We give an example to illustrate these results.Example 23.
Letα be a generator of the multiplicative group of F4; that is, α is a zero of z2+z+1∈F2[z]. Let Θ be the automorphism a↦a2 of F4. To list all [4,1] skew Θ-cyclic codes over F4, we find all monic degree 1 right factors of x4-1∈F4[x;Θ]. They are (18)f1=x+1,f2=x+α,f3=x+α2.Similarly, to list all [4,2] skew Θ-cyclic codes over F4, we find all monic degree 2 right factors of x4-1∈F4[x;Θ]. They are (19)g1=x2+1,g2=x2+αx+α2,g3=x2+α2x+α,g4=x2+α2x+α2,g5=x2+x+α,g6=x2+x+α2,g7=x2+αx+α.Letg be a right divisor of xn-1 of degree r. Then the skew Θ-cyclic code is [n,n-r] linear code with generator matrix (20)G≔g0g1⋯gr0⋯00Θg0⋯Θgr-1Θgr⋯0⋮⋮⋮⋮⋮⋮⋮00⋮Θn-r-1g0Θn-r-1g1⋮Θn-r-1gr.A right factor of degreen-k of xn-1 generates a linear code with parameters (n,k). If Θ is not the identity, then the skew polynomial ring Fpm[x;Θ] is in general not a unique factorization. In this case, we have more right factors than in the commutative case. For small values of n, all right skew factors of xn-1 can be found by a computational algebra system such as MAGMA (cf. [30]). Minimum distance of a code can be also calculated by using the MAGMA procedures. However, these procedures must be spent a long time for larger codes to check them. Therefore, the process will only find the smaller codes. The code parameters and the number of codes are introduced with these parameters (n,k,dmin) because many different codes with the same minimum distance can be found. A generating polynomial for one code respected the class of parameters (n,k,dmin) is also exhibited. Table 1, computed by Bosma et al. [30], provides parameters and generating polynomials of skew Θ-cyclic codes over F4, where Θ is the Frobenius automorphism and α is a generator of the multiplicative group of F4.Table 1
Generating polynomial of skewΘ-cyclic codes over F4 with their parameters (n,k,dmin), where n≤56. This result is given by Boucher et al. [3].
(
n
,
k
,
d
m
i
n
)
Number
g
(
30,16,9
)
422
x
14
+
x
13
+
α
x
11
+
x
10
+
x
9
+
x
8
+
α
x
7
+
x
6
+
α
x
5
+
α
2
x
4
+
α
2
x
2
+
α
x
+
α
2
(
36,20,10
)
13
x
16
+
α
2
x
15
+
x
13
+
α
2
x
12
+
x
11
+
α
x
10
+
x
9
+
α
2
x
8
+
α
x
7
+
α
x
6
+
α
x
4
+
α
2
x
3
+
α
2
x
2
+
1
(
40,16,15
)
6
x
24
+
α
x
23
+
x
22
+
x
21
+
α
2
x
20
+
α
x
19
+
α
x
18
+
α
x
17
+
x
15
+
x
14
+
x
13
+
α
x
11
+
α
2
x
10
+
x
9
+
x
8
+
x
7
+
α
2
x
6
+
α
x
5 +α2x4+αx2+α2
(
42,23,11
)
92
x
19
+
x
17
+
α
2
x
16
+
α
x
15
+
α
2
x
14
+
α
x
13 + αx11 + α2x10+αx9+x7+αx6+α2x5+αx4+αx+α2
(
42,17,16
)
3
x
25
+
x
23
+
α
x
22
+
x
21
+
x
20
+
x
19
+
x
18
+
α
2
x
17
+
α
2
x
16
+
α
x
15
+
α
x
14
+
x
13
+
x
11
+
x
10
+
x
8
+
α
2
x
4
+
α
2
x
3
+
x
2 +αx+1
(
48,25,13
)
2
x
23
+
α
2
x
22
+
x
21
+
α
x
20
+
α
x
19
+
α
2
x
18
+
α
x
17
+
α
x
14
+
α
2
x
13
+
α
2
x
11
+
x
9
+
α
x
7
+
x
6
+
x
3
+
α
2
x
2
+
1
(
48,19,17
)
2
x
29
+
α
2
x
28
+
x
26
+
α
x
25
+
α
2
x
24
+
α
x
23
+
α
x
21
+
α
x
20
+
α
2
x
19
+
α
x
18
+
α
x
17
+
α
x
16
+
x
15
+
x
14
+
α
x
13
+
α
x
10
+
α
x
8 +α2x7+x6+x5+x4+α2x3+x2+α2
(
56,30,14
)
1
x
26
+
x
23
+
α
x
22
+
α
2
x
21
+
α
x
20
+
α
2
x
19
+
α
2
x
18
+
α
x
17
+
x
16
+
x
14
+
x
13
+
α
x
11
+
α
2
x
10
+
α
2
x
9
+
α
2
x
8
+
α
x
7
+
α
2
x
6 +αx5+αx5+α2x4+x2+α2x+α2Givenn-tuples x=(x0,x1,…,xn-1) and y=(y0,y1,…,yn-1)∈Fpm, their inner product or dot product is defined in the usual way: (21)x∘y=x0y0+x1y1+⋯+xn-1yn-1,evaluated in Fpm. Two codewords x,y are calledorthogonal if x∘y=0. For a linear code C over Fpm, its dual code C⊥ is the set of n-tuples over Fpm that are orthogonal to all codewords of C; that is, (22)C⊥=x∣x∘y=0,∀y∈C.A code C is calledself-orthogonal if C⊂C⊥, and it is calledself-dual if C=C⊥. The following result is well known (cf. [29]).Lemma 24 (see [29, Corollary 18]).
LetΘ be an automorphism of Fpm, n an integer divisible by the order of Θ, and λ a unit in Fpm which is fixed by Θ. Let g(x)=∑i=0r-1‍gixi+xr and h(x)=∑i=0n-r-1‍hixi+xn-r such that h(x)g(x)=xn-1. The dual of the skew Θ-cyclic code generated by g(x) in Fpm[x;Θ]/xn-1 is the skew Θ-cyclic code generated by (23)g⊥=1+Θhn-r-1x+⋯+Θn-rh0xn-r.We give an example to illustrate how we use Lemma24 to determine Euclidean self-dual Θ-cyclic codes.Example 25.
Letα be a generator of the multiplicative group of F4; that is, α is a zero of z2+z+1∈F2[z]. Let Θ be the automorphism a↦a2 of F4. We find all Euclidean self-dual Θ-cyclic codes over F4 in F4[x;Θ]/(x4-1). From Example 23, we can list all monic degree 2 right factors of x4-1∈F4[x;Θ]. Put hi(i={1,…,7}) to be all monic degree 2 right factors such that hi·gi=x4-1, ∀i∈{1,…,7}. Then we have (24)h1=x2+1,h2=x2+αx+α,h3=x2+α2x+α2,h4=x2+α2x+α,h5=x2+x+α2,h6=x2+x+α,h7=x2+αx+α2.Applying Lemma 24, we have (25)g1⊥=x2+1,g2⊥=αx2+α2x+1,g3⊥=α2x2+αx+1,g4⊥=αx2+αx+1,g5⊥=α2x2+x+1,g6⊥=αx2+x+1,g7⊥=α2x2+α2x+1,where the dual of the skew Θ-cyclic code generated by gi(x) in F4[x;Θ]/xn-1 is the Θ-cyclic code generated by gi⊥. Suppose that Ci, the skew Θ-cyclic code generated by gi(x) in F4[x;Θ]/xn-1, is an Euclidean self-dual Θ-cyclic code. Then we have C=C⊥. This implies that gi⊥ is a constant multiple of gi. From this, the skew Θ-cyclic codes generated by g1(x),g2(x),g3(x) are Euclidean self-dual Θ-cyclic codes.We now turn our attention to Euclidean self-dualΘ-cyclic codes over F4 (cf. [29]). Suppose that Θ is the Frobenius automorphism and α is a generator of F4. It is easy to see that n must be an even number. In fact, by Lemma 24, if n is odd, then there are no Euclidean self-dual codes. Therefore, n=2r for some r∈Z+. Let C be a self-dual code. Applying Lemma 24, the coefficients of the generating polynomial g⊥ of C⊥ is expressed. Since C=C⊥, g⊥ and g must differ by a constant multiple and deg(g)=deg⁡(g⊥). Now assume that g=∑i=0r-1‍gixi+xr with g0≠0 is the generator polynomial of the self-dual Θ-cyclic code C. Assume that h=xr+∑i=0r-1‍hixi such that gh=xn-1. From Lemma 24, the code C⊥ is generated by g⊥=1+∑i=1r‍Θi(hr-i)xi. Since g⊥ is a constant multiple of g,g⊥=Θr(h0)g. This implies that if the coefficients of both polynomials g and g⊥ are compared, then system (26) is built as follows:(26)1=Θrh0g0,Θihr-i=Θrh0gi,i=0,1,…,r-1.Since Θ2(α)=α, it is easy to see that Θ=Θ-1. By assumption, g0≠0. This implies that g0-1=g02. Hence, system (26) becomes(27)h0=Θrg02,hr-i=Θig02·gi,i=0,1,…,r-1.System (27) allows expressing the coefficients of h as follows: (28)h=Θ2g02+∑i=1rΘr-ig02Θr-igr-ixi.From Θi(a)=a(2imod⁡3), (28) becomes (29)h=g02r-1mod⁡3+∑i=1rg02r-i+1mod⁡3gr-i2r-imod⁡3xi,where powers of gi are of degree less than 4. By using the rule xia=a(2imod⁡3)xi and expanding the skew product (xn-1)-(h·g)=0, (2r+1) polynomial equations in the coefficients gi of degree less than 4 in each variable can be determined. From gi∈F4, gi4-gi=0 for any i=1,…,r. Adding r equations gi4-gi=0 to 2r+1 polynomial equations in r variables of degree less than 4 to have a system, then the solutions of this system can be found by using Groebner bases in MAGMA system because the solution set must be finite. This shows that all polynomials g corresponding to a solution will be listed and hence the linear code which it generates and its minimum Hamming distance can be computed. Then all Euclidean self-dual Θ-cyclic codes of length n≤40 in F4[x;Θ] will be exhibited. In [29], Boucher and Ulmer gave the table of Euclidean self-dual codes over F4 and n≤40. We refer the readers to [29] for more details.Recall that Hermitian inner product is denoted and calculated byu,vH=∑i=0n-1‍uiΘ(vi), for all u=(u0,…,un-1) and v=(v0,…,vn-1) in Fpmn. Then we give the definition of the Hermitian dual code of a code C as follows: (30)C⊥H=v∈Fpmn∣v,cH=0,∀c∈C.If C=C⊥H, then C is said to be a Hermitian self-dual code. It is easy to check that if n is odd, then there is no Hermitian self-dual of Θ-cyclic codes. Therefore, n must be an even number. Suppose that the order m of Θ divides n=2r. Let g=∑j=0r‍gjxj and h=∑i=0r‍hixi be elements of Fpm[x;Θ] such that h·g=x2r-1. The Hermitian dual of a Θ-cyclic code generated by g in Fpm[x;Θ]/x2r-1 is again Θ-cyclic code and is generated by gH=Θ(hr)+Θ2(hr-1)x+⋯+Θr+1(h0)xr [29, Lemma 21]. Similar to the case of Euclidean self-dual codes, all the Hermitian self-dual Θ-cyclic codes of length n≤40 in F4[x;Θ] can be found. The polynomial h of Hermitian self-dual code in F4[x;Θ] can be expressed as follows: (31)h=xr+∑i=1r-1Θr-i+1g02Θr-i+1gr-ixi+Θr+1g02.In this case, the coefficient of (31) is shifted by Θm-1=Θ. Expanding the skew product x2r-1-h·g=0 which gives again a polynomial system of equations, the solutions of this system can be also computed by using Groebner bases in MAGMA because the solution set must be finite. Similar to the case of Euclidean self-dual codes, in [29], Boucher and Ulmer also gave the table of Hermitian self-dual codes over F4 and n≤40. For more details we refer the readers to [29].


## 4. Decoding SkewΘ-Cyclic Codes over Finite Fields

In coding theory, BCH codes were invented in 1959 by French mathematician Alexis Hocquenghem and independently in 1960 by Raj Bose and D. K. Ray-Chanahuri. General decoding procedure for decoding BCH codes with designed distance is introduced in [31]. In this section, we first give the algorithm for decoding with cyclic codes in Fpm[x]. After that, we will modify the algorithm for decoding skew BCH codes.LetC be [n,k,d] cyclic code over Fpm with generator polynomial g(x) of degree n-k. Suppose that c(x)∈C is transmitted and y(x)=c(x)+e(x) is received, where e(x)=e0+e1x+⋯+en-1xn-1 is the error vector with wt(e(x))≤t and t=(d-1)/2. Let Rg(x) be the unique remainder when h(x) is divided by g(x) according to the Division Algorithm; that is, Rg(x)(h(x))=r(x), where h(x)=g(x)f(x)+r(x), with r(x)=0 or deg⁡r(x)<n-k. Then the function Rg(x) satisfies the following properties.Theorem 26 (see [27, Theorem 4.6.1]).
With the preceding notation the following statements hold:(i)
Rg(x)(ah(x)+bh′(x))=aRg(x)(h(x))+bRg(x)(h′(x)) for all h(x); h′(x)∈Fpm[x] and all a;b∈Fpm,(ii)
Rg(x)(h(x)+a(x)(xn-1))=Rg(x)(h(x)),(iii)
Rg(x)(h(x))=0 if and only if h(x)(mod⁡(xn-1))∈C,(iv)
Ifc(x)∈C, then Rg(x)(c(x)+e(x))=Rg(x)(e(x)),(v)
IfRg(x)(e(x))=Rg(x)(e′(x)), where e(x) and e′(x) each have weight at most t, then e(x)=e′(x),(vi)
Rg(x)(h(x))=h(x) if deg⁡h(x)<n-k.Theorem 27 (see [27, Theorem 4.6.2]).
Letg(x) be a monic divisor of xn-1 of degree n-k. If Rg(x)(h(x))=s(x), then Rgx(xh(x)(mod⁡(xn-1))=Rg(x)(xs(x))=xs(x)-g(x)sn-k-1, where sn-k-1 is the coefficient of xn-k-1 in s(x).Define the syndrome polynomialS(h(x)) of any h(x) to be (32)Shx=Rgxxn-khx.We now describe the first version of the Meggitt Decoding Algorithm and we provide an example to illustrate each step.Step 1.
Find the syndrome polynomialS(e(x)) of error patterns e(x)=∑i=0n-1‍eixi such that wt(e(x))≤t and en-1≠0.Example 28 (see [27, Example 4.6.3]).
LetC be the [15,7,5] binary cyclic code with a generating polynomial g(x)=1+x4+x6+x7+x8 = (x-ξ)(x-ξ2)(x-ξ3)(x-ξ4)(x-ξ6)(x-ξ8)(x-ξ9)(x-ξ12), where ξ is a 15th root of unity in F16. Then the syndrome polynomial of e(x) is S(e(x))=Rg(x)(x8e(x)). The syndrome polynomial for polynomial 1+x14 can be computed as follows. First, it is easy to see that Rg(x)(x8)=1+x4+x6+x7. Then S(1+x14)=Rg(x)(x8(1+x14))=Rg(x)(x8)+Rg(x)(x7)=1+x4+x6. Applying Theorem 26, Rg(x)(x9)=Rg(x)(xx8) = Rg(x)(x+x5+x7)+Rg(x)(x8) = x+x5+x7+1+x4+x6+x7 = 1+x+x4+x5+x6. Similarly, by applying Theorems 26 and 27, all syndrome polynomials will be determined. Table 2 shows all syndrome polynomials.Table 2
e
(
x
)
S
(
e
(
x
)
)
e
(
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7Step 2.
Assume thaty(x) is the received polynomial. Then the syndrome polynomial S(y(x))=Rg(x)(xn-ky(x)) can be computed. Applying Theorem 26(iv), S(y(x))=S(e(x)), where y(x)=c(x)+e(x) and c(x)∈C.Example 29 (see [27, Example 4.6.4]).
We give an example for Step2. We continue Example 28. Suppose that y(x)=1+x4+x7+x9+x10+x12 is a received polynomial. This implies that S(y(x))=x+x2+x6+x7.Step 3.
IfS(y(x)) is in list computed in Step 1, then the error polynomial e(x) can be computed and it can be subtracted from y(x) to the corrected codeword c(x)=y(x)-e(x). If S(y(x)) is not appearing in the list computed in Step 1, then the process will continue to Step 4.Step 4.
It is continuing to compute the syndrome polynomial ofxy(x),x2y(x),… until the syndrome polynomial is in the list from Step 1. If S(xiy(x)) is in this list and it is associated with the error polynomial e∗(x), then the received vector is decoded as y(x)-xn-ie∗(x). By using Theorem 27, Rg(x)(xn-ky(x))=S(y(x))=∑i=0n-k-1‍sixi and S(xy(x))=Rg(x)(xn-kxy(x)) = Rg(x)(x(xn-ky(x))) = Rg(x)(xS(y(x))) = xS(y(x))-sn-k-1g(x).We finish this part by the following example.Example 30 (see [27, Example 4.6.6]).
We can see thatS(y(x))=x+x2+x6+x7 is not in the list computed in Step 1, then we continue to compute S(xy(x))=x(x+x2+x6+x7)-1·g(x)=1+x2+x3+x4+x6, which is not also appearing in the list in Step 1. It is easy to check that S(x2y(x))=x(1+x2+x3+x4+x6)-0·g(x)=x+x3+x4+x5+x7 is in the list in Step 1. This implies that y(x) is decoded as y(x)-(x2+x12)=1+x2+x4+x7+x9+x10.Suppose thatα∈Fq is a primitive (q-1)th root of unity, n is even, q=2m, and Θ is an automorphism of Fq such that Θ(α)=α2. We give two results in [3] and use them later.Lemma 31 (see [3, Proposition 1]).
ForP=∑k=0n-1‍akxk∈Fq[x;Θ], β∈Fq and r∈Fq the remainder of the right division of P by x-β, then r=P¯ is a (classical) polynomial given by P¯=∑k=0n-1‍akz2k-1∈Fq[z].Lemma 32 (see [3, Proposition 2]).
Letn be even, q=2n, and α a primitive (q-1)th root of unity. Let C be a Θ-cyclic code with Θ(α)=α2. Let G∈Fq[x;Θ] be its generating polynomial such that G is a right divisor of xn-1 in Fq[x;Θ] and x-αk is a right factor of G for k∈{1,…,d-1}. The distance of the code C is equal to its designed distance d.We now introduce the procedure for decoding skew BCH codes. Assume thate=ei1xi1+⋯+eirxir is the error polynomial with i1<i2<⋯<ir, where r≤(d-1)/2. The polynomial Sd(z)=∑k=1d-1‍Rem(e,x-αk)zk-1∈Fpm[z] is called asyndrome polynomial of e. Note that Rem(e,x-αk) is to be computed in the skew polynomial Fpm[x;Θ]. Hence, (33)Sdz=∑k=1d-1Reme,x-αkzk-1=∑k=1d-1e¯αkzk-1,where e¯(z)=∑k=1r‍eikzjk∈Fpm[z] and jk=2ik-1. The polynomials σ(z)=∏k=1r‍(1-αjkz) and w(z)=∑l=1r‍eilαjl-∏k≠l‍(1-αjkz) are calledpseudolocator polynomial andevaluator polynomial, respectively. Let (34)Sz=∑k=1∞e¯αkzk-1=Sdz+zd-1∑k=0∞e¯αk+1+dzk.This implies that σ(z)S(z)=w(z). This equation can be written to become σ(z)+Sd(z)+v(z)zd-1=w(z), where v(z)=σ(z)∑k=1∞‍e¯(αk+1+d)zk.Applying the Euclidean Algorithm to the polynomialsSd(z) and zd-1 in Fpm[z], three sequences (ri(z)),(ui(z)), and (vi(z)) are defined as follows: (35)r-1z=zd-1,r0=Sdz,u-1z=0,u0z=1,v-1z=1,v0z=0and ri(z)=ri-2(z)-qi(z)ri-1(z), ui(z)=ui-2(z)-qi(z)ui-1(z), and vi(z)=vi-2(z)-qi(z)vi-1(z) with deg⁡(ri(z))<deg⁡(ri-1)(z). The process will stop whenever k can be determined satisfying deg⁡(rk-1)≥(d-1)/2 and deg⁡(rk)<(d-1)/2. From this, rk(z), σ(z), and w(z) can be computed by three equations as follows: (36)ukzSdz+vkzzd-1=rkz;σz=ukzuk0;wz=rkzrk0.From the roots of the pseudolocator polynomial σ(z), all jl, l∈{1,2,…,r}, will be listed. This shows that (37)eil=α-jlwα-jl∏k≠l1-αjk-jl,l∈1,2,…,r.From the equation above, all coefficients of e are also determined. For each jl, a finite number of possibilities il solutions to the equation jl≡2il-1(mod⁡n) can be found. Similarly to the procedure for decoding BCH codes, this process will test until the skew polynomial e is determined. Since e is unique, the decoded word can be exhibited, as required.We conclude this section by an example provided by Boucher et al. in [3] to illustrate this process in detail.Example 33 (see [3]).
Letα be such that α210-1=1. Suppose that m=n=10. Then the polynomial g(x)=x6+α345x5+α643x4+α878x3+α670x2+α1020x+α777 is a divisor of x10+1 in F210[x;Θ]. This implies that g(x) is the generator polynomial of a Θ-cyclic code of length 10 over F210. We can see that x-αk is a right factor of g(x) for all k∈{1,…,6}. Hence, the designed distance of the code C is d=7. Now we consider f(x)=α654x9+α547x8+α650x7+α16x6+α567x5+α29x4+α87x3+α696x2+α252x+α555 and an error e=α341x9+α682x8+α682. The pertubed codeword h is (38)f+e=α818x9+α775x8+α650x7+α16x6+α567x5+α29x4+α87x3+α696x2+α252x+α557.Since d=7 and polynomial h, we have the syndrome polynomial (39)S7z=α404z5+α403z4+α601z3+α645z2+α614z+α406.Applying Euclid Algorithm to S7(z) and z6 in F210[z] with (d-1)/2=3, we can get the pseudolocator polynomial σ(z)=α766z3+α642z2+α241z+1 and the evaluator polynomial w(z)=α84z2+α185z+α406. The roots of the polynomial σ(z) are 1, α512, and α768. From this, we have r=3, j1=0, j2=511, and j3=255. By the polynomial w(z), we can find ei1=α682, ei2=α341, and ei3=α682. Combining this result and the equations 2ik-1≡jk(mod⁡10), we have i1=0(mod⁡10), i2=1,5,9(mod⁡10), and i3=4,8(mod⁡10). Then we can list all possible errors as follows: (40)α341x9+α682x8+α682;α682x8+α341x5+α682;α341x5+α682x4+α682;α682x8+α341x+α682;α341x9+α682x4+α682;α682x4+α341x+α682.It is easy to find that e=α341x9+α682x8+α682.


## 5. SkewΘ-λ-Constacyclic Codes over Finite Chain Rings

Constacyclic codes have practical applications as they can be efficiently encoded using simple shift registers. They have rich algebraic structures for efficient error detection and correction, which explains their preferred role in engineering. Classically, the algebraic structures of constacyclic codes are determined by ideals in the polynomial rings over finite fields, Galois rings, and finite chain rings. In [3], Boucher et al. generalized the notion of cyclic codes by using generator polynomial in noncommutative skew polynomial rings. Since there are much more skew cyclic codes, the new class of codes allowed them to systematically search for codes. Later on, the approach has been extended to codes over Galois rings [29]. In 2012, Jitman et al. [26] studied skew Θ-λ-constacyclic codes over finite chain rings. These codes have been studied for a particular case when codes are generated by monic right divisors of xn-λ, where λ is a unit in the finite chain rings fixed by a given automorphism. Similarly to the case of skew Θ-λ-constacyclic codes over finite fields, when Θ is the identity automorphism, they become classical constacyclic codes over finite chain rings. Therefore, skew Θ-λ-constacyclic codes over finite chain rings can be considered as a generalization of classical constacyclic codes over finite chain rings. This is the reason why the study of skew Θ-λ-constacyclic codes over finite chain rings is important. In this section, we overview the study of skew Θ-λ-constacyclic codes over finite chain rings studied by Jitman et al. [26].A finite commutative ring with identity is called afinite chain ring if its ideals are linearly ordered by inclusion or, equivalently, its ideals are principal and its maximal ideal is unique. In [32], it is known that a finite chain ring is local and its unique maximal ideal is principal. Constacyclic codes over a finite commutative chain ring have been studied by many authors (see, e.g., [23, 33–36]). The structure of constacyclic codes is also introduced over a special family of finite chain rings of the form Fpm+uFpm. Recently, skew Θ-codes over finite fields and Galois rings were studied by Boucher et al. Motivated by these results, in [26], Jitman et al. generalized the concept of skew Θ-λ-constacyclic codes over finite fields and Galois rings to that over finite chain rings. The structure of all skew Θ-λ-constacyclic codes over a finite chain ring is determined. Moreover, Euclidean and Hermitian dual codes of skew Θ-cyclic and negacyclic codes are considered. They also studied skew Θ-λ-constacyclic codes over a special case Fpm+uFpm of a finite chain ring.In this section, letR be a finite chain ring with unique maximal ideal γ. Then γ is a nilpotent ideal of R and we denote its nilpotency index by t. Hence, the ideals of R form the following chain: (41)R=1⊋γ⊋⋯⊋γt-1⊋γt=0.Analogous to the case of finite fields, the set of automorphisms ofR forms a group under composition, denoted by Aut(R). Many classes of finite chain rings have nontrivial automorphism groups. For examples, Aut(GR(pe,m)) is nontrivial if and only if m≥2 (cf. [16]) and Aut(Fpm+uFpm+⋯+ue-1Fpm) is nontrivial if and only if m≥2 or p is odd or e≥3 (cf. [37, Proposition 1]).We know thatFpm[x;Θ] is left and right Euclidean ring whose left and right ideals are principal. Unlike the ring Fpm[x;Θ], if R is a finite chain ring, then the skew polynomial ring R[x;Θ] is neither left nor right Euclidean ring. Therefore, we need to define left and right divisions. Suppose that f(x)=∑i=0s‍aixi and g(x)=∑j=0t‍bjxj, where bt is a unit in R and s⩾t. We can see that the degree of polynomial (42)fx-as·Θs-tbt-1xs-tgxis less than the degree of f(x). By the inductive method, we can obtain skew polynomials q(x) and r(x) such that f(x)=q(x)g(x)+r(x) with deg⁡(r(x))<deg⁡(g(x)) or r(x)=0. If r(x)=0, then we say that g(x) is a right divisor of f(x). The skew polynomials q(x) and r(x) are unique. They are calledthe right quotient andright remainder, respectively. Note that if s<t, then we put f(x)=0·g(x)+f(x). This algorithm is called theRight Division Algorithm in R[x;Θ]. TheLeft Division Algorithm in R[x;Θ] can be defined similarly, using the fact that the degree of (43)fx-gxΘ-tasbt-1xs-tis less than the degree of f(x). Now we recall the definition of skew Θ-λ-constacyclic codes in R[x;Θ]. Given an automorphism Θ of R and a unit in R, a linear code C is said to be skew Θ-λ-constacyclic if C is closed under the Θ-λ-constacyclic shift τΘ,λ:Rn→Rn defined by (44)τΘ,λc0,c1,…,cn-1=Θλcn-1,Θc0,…,Θcn-2.


### 5.1. SkewΘ-λ-Constacyclic Codes over Finite Chain Rings

For a skew polynomialg(x) in R[x;Θ], then a left ideal generated by g(x), denoted by gx, is in general not a two-sided ideal. However, if g(x)=xth(x)(t∈N0) such that h(x) is central (i.e., commutes with all elements of R[x;Θ]), then 〈f(x)〉 is a principal two-sided ideal in R[x;Θ]. From this remark, the following corollary is a direct consequence.Corollary 34 (see [26, Corollary 2.2]).
Iff(x) is a monic central skew polynomial of degree n, then the skew polynomials of degree less than n are canonical representatives of the elements in R[x;Θ]/〈f(x)〉.Analogous to classical constacyclic codes, we study skewΘ-λ-constacyclic codes as left ideals in R[x;Θ]/xn-λ. Note that R[x;Θ] is a noncommutative ring. So we need to have the conditions of Θ and λ which ensure that xn-λ is a two-sided ideal.Lemma 35 (see [26, Proposition 2.2]).
Letn be a positive integer and λ a unit in R. Then the following statements are equivalent: (i)
xn-λ is central in R[x;Θ].(ii)
xn-λ is a two-sided ideal.(iii)
n is a multiple of the order of Θ and λ is fixed by Θ.ForΘ-λ-constacyclic codes over finite fields, a code C is a skew Θ-λ-constacyclic code if and only if C is a left ideal 〈g(x)〉⊆Fpm[x;Θ]/xn-λ, where g(x) is right divisor of xn-λ. In the case finite chain rings, the following theorem is analogous to that for Θ-λ-constacyclic codes over finite fields.Theorem 36 (see [26, Theorem 2.2]).
LetΘ be an automorphism of R, n an integer divisible by the order of Θ, and λ a unit in R which is fixed by Θ. Then the code C is a skew Θ-λ-constacyclic code if and only if C is a left ideal 〈g(x)〉⊆R[x;Θ]/〈xn-λ〉, where g(x) is a right divisor of xn-λ.From this theorem, we can find a skewΘ-λ-constacyclic code as a left ideal 〈g(x)〉⊆R[x;Θ]/xn-λ, where g(x) is a right divisor of xn-λ. However, it is not easy to list all skew Θ-λ-constacyclic codes because R[x;Θ] is not unique factorization ring. Therefore, there are many more right factors than in the commutative case, which in turn produces many more skew Θ-λ-constacyclic codes.Example 37.
LetR=F3+uF3 be a finite chain ring. We consider the automorphism Θid,2 of F3+uF3, where Θid,2(a+ub)=a+2bu. Then we have two irreducible factorizations of x6-1 in (F3+uF3)[x;Θid,2]: (45)x6-1=x+13x+23=x2+ux+23.Given a monic right divisor of degreen-k of xn-λ:g(x)=∑i=0n-k-1‍gi(x)+xn-k, then a generator matrix of the Θ-λ-constacyclic code C generated by g(x) is given by (46)G≔g0⋯gn-k-110⋯00Θg0⋯Θgn-k-11⋯00⋯⋯⋯Θ2gn-k-1⋯0⋮⋮⋮⋮⋮⋮⋮0⋯0Θk-1g0⋯Θk-1gn-k-11.The rows of G are linearly independent. Then we have the following result.Proposition 38 (see [26, Proposition 3.1]).
Letg(x) be a right divisor of xn-λ. Then the Θ-λ-constacyclic code C generated by g(x) is a free R-module with |C|=|R|n-deg⁡(g(x)).Similarly, in the case of finite fields, we denoteRΘ, the subring of R fixed by Θ. Then we have the following result.Proposition 39 (see [26, Proposition 3.2]).
Letg(x) be a monic right divisor of xn-λ in R[x;Θ]. The skew Θ-λ-constacyclic code generated by g(x) is λ-constacyclic if and only if g(x)∈RΘ[x;Θ].LetC be a Θ-λ-constacyclic code. In the following lemma, the parity-check matrix for C is introduced.Lemma 40 (see [26, Proposition 3.3]).
LetC be the Θ-λ-constacyclic code generated by a monic right divisor g(x) of xn-λ and h(x)≔(xn-λ)/g(x). Then the following statements hold: (i)
Forc(x)∈R[x;Θ], c(x)∈C if and only if c(x)h(x)=0 in R[x;Θ]/xn-λ.(ii)
Ifh(x)=∑i=0k-1‍hixi+xk, then the matrix (47)H≔1Θhk-1⋯Θkh00⋯001Θ2hk-1⋯Θk+1h0⋯000⋯⋯⋯⋯0⋮⋮⋮⋮⋮⋮⋮00⋯1Θn-khk-1⋯Θn-1h0is a parity-check matrix forC.In the next part, we study the Euclidean and Hermitian dual codes of skewΘ-λ-constacyclic codes over finite chain rings. Suppose that the length n of codes is divisible by the order of Θ, and λ is a unit in R which is fixed by Θ. Euclidean inner product is defined by u,v=∑i=0n-1‍uivi, for u=(u0,u1,…,un-1) and v=(v0,v1,…,vn-1) in Rn. In special case, if the order of Θ is 2, then we can also give the Hermitian inner product, denoted by u,vH=∑i=0n-1‍uiΘ(vi). If u,v=0 (resp., u,vH=0), then u and v are called Euclidean orthogonal (resp., Hermitian orthogonal). The Euclidean and Hermitian dual code of a code C are defined to be (48)C⊥=v∈Rn∣v,c=0,∀c∈C,(49)C⊥H=v∈Rn∣v,cH=0,∀c∈C,respectively. If C=C⊥ (C=C⊥H), then C is said to be Euclidean (Hermitian) self-dual code. We get a main result which describes the relationship between a skew Θ-λ-constacyclic code and its dual.Lemma 41 (see [26, Lemma 3.1]).
LetC be a code of length n over R. Then C is skew Θ-λ-constacyclic if and only if C⊥ is Θ-λ-1-constacyclic. In particular, if λ2=1, then C is Θ-λ-constacyclic if and only if C⊥ is Θ-λ-constacyclic.


### 5.2. Euclidean Dual Codes

We denote thatR[x;Θ]S-1 is the right localization of R[x;Θ]. The following theorem will discuss the necessary and sufficient conditions for R[x;Θ] to have the right localization.Theorem 42 (see [26, Theorem 2.1]).
LetS={xi∣i∈N}. Then R[x;Θ] has the right localization at S if and only if both the following conditions hold: (i)
For allxi∈S and a(x)∈R[x;Θ], there exist xj∈S and b(x)∈R[x;Θ] such that a(x)xi=xjb(x).(ii)
Givena(x)∈R[x;Θ] and xi∈S, if xia(x)=0, then there exists xj∈S such that a(x)xj=0.Before determining the structure of dual codes, we get the following result.Lemma 43 (see [26, Proposition 2.4]).
Letφ:R[x;Θ]→R[x;Θ]S-1 defined by (50)φ∑i=0kaixi=∑i=0kx-iai.Then φ is a ring antimonomorphism.From Lemma41, it is easy to verify that the Euclidean dual of a skew Θ-λ-constacyclic code C is again a skew Θ-λ-constacyclic code. We introduce a result about Euclidean dual codes. To do that, we need the following lemma.Lemma 44 (see [26, Lemma 3.2]).
Assume thatλ2=1. Let a(x)=a0+a1x+⋯+an-1xn-1 and b(x)=b0+b1x+⋯+bn-1xn-1 be polynomials in R[x;Θ]. Then the following statements are equivalent: (i)
The coefficient vector ofa(x) is Euclidean orthogonal to the coefficient vector of xi(xn-1φ(b(x))) for all i∈{0,1,…,n-1}, where φ:R[x;Θ]→R[x;Θ]S-1 is a ring antimonomorphism defined in Lemma 43.(ii)
(a0,a1,…,an-1) is Euclidean orthogonal to (bn-1,Θ(bn-2),…,Θn-1(b0)) and all its Θ-λ-constacyclic shifts.(iii)
a(x)b(x)=0 in R[x;Θ]/xn-λ.Theorem 45 (see [26, Theorem 3.3]).
Assume thatλ2=1. Let g(x) be a right divisor of xn-λ and h(x)≔(xn-λ)/g(x). Let C be the Θ-λ-constacyclic code generated by g(x). Then the following statements hold: (i)
The skew polynomialxdeg⁡(h(x))φ(h(x)) is a right divisor of xn-λ.(ii)
The Euclidean dualC⊥ is a Θ-λ-constacyclic code generated by xdeg⁡(h(x))φ(h(x)).Theorem 46 (see [26, Theorem 3.4]).
Assume thatλ2=1 and n is even, denoted by n=2k. Let g(x)=∑i=0k-1‍gixi+xk be a right divisor of xn-λ. Then the Θ-λ-constacyclic code generated by g(x) is Euclidean self-dual if and only if (51)∑i=0k-1gixi+xkΘ-kg0-1+∑i=0k-1Θi-kg0-1gk-ixi+xk=xn-λ.Theorem46 provided the necessary and sufficient conditions for a skew Θ-λ-constacyclic code to be Euclidean self-dual code. By applying Theorem 46, we can see that if the order of Θ divides k and λ≠-1, then there are no Euclidean self-dual skew constacyclic codes of length 2k. Moreover, if Θ is the identity automorphism and λ≠-1, then there are no Euclidean self-dual codes.


### 5.3. Hermitian Dual Codes

The Hermitian inner product is defined only when the order ofΘ is 2. Therefore, in this subsection, we always suppose that the order of Θ is 2. We first have some characterizations of Hermitian duality.Lemma 47 (see [26, Lemma 3.5]).
LetC be a code of length n over R. Then C is skew Θ-λ-constacyclic if and only if C⊥H is Θ-λ-1-constacyclic. In particular, if λ2=1, then C is Θ-λ-constacyclic if and only if C⊥H is Θ-λ-constacyclic.Letϕ be a ring automorphism of R[x;Θ] defined by ϕ(∑i=0s‍aixi)=∑i=0s‍Θ(ai)xi. Then we have the following result.Lemma 48 (see [26, Lemma 3.6]).
Assume thatλ2=1. Let a(x)=a0+a1x+⋯+an-1xn-1 and b(x)=b0+b1x+⋯+bn-1xn-1 be polynomials in R[x;Θ]. Then the following statements are equivalent: (i)
The coefficient vector ofa(x) is Euclidean orthogonal to the coefficient vector of xiϕ(xn-1φ(b(x))) for all i∈{0,1,…,n-1}, where φ:R[x;Θ]→R[x;Θ]S-1 is a ring antimonomorphism defined in Lemma 43.(ii)
(a0,a1,…,an-1) is Hermitian orthogonal to (Θ-1(bn-1),(bn-2),…,Θn-2(b0)) and all its Θ-λ-constacyclic shifts.(iii)
a(x)b(x)=0 in R[x;Θ]/〈xn-λ〉.Theorem 49 (see [26, Theorem 3.7]).
Assume thatλ2=1. Let g(x) be a right divisor of xn-λ and h(x)≔(xn-λ)/g(x). Let C be the Θ-λ-constacyclic code generated by g(x). Then the following statements hold: (i)
The skew polynomialϕ(xdeg(h(x))φ(h(x))) is a right divisor of xn-λ.(ii)
The Hermitian dualC⊥H is a Θ-λ-constacyclic code generated by ϕ(xdeg⁡(h(x))φ(h(x))).Similar to the case of the Euclidean self-dual code, we have the necessary and sufficient conditions for aΘ-λ-constacyclic code to be Hermitian self-dual.Theorem 50 (see [26, Theorem 3.8]).
Assume thatλ2=1 and n is even, denoted by n=2k. Let g(x)=∑i=0k-1‍gixi+xk be a right divisor of xn-λ. Then the Θ-λ-constacyclic code generated by g(x) is Hermitian self-dual if and only if (52)∑i=0k-1gixi+xkΘ-k-1g0-1+∑i=0k-1Θi-k-1g0-1gk-ixi+xk=xn-λ.From this theorem, ifk is odd and λ≠-1, then there are no Hermitian self-dual Θ-λ-constacyclic codes of length 2k.


### 5.4. Skew Constacyclic Codes overFpm+uFpm

The class of finite chain rings of the formFpm+uFpm has been used widely as alphabets in certain constacyclic codes. It has been studied by many researchers (see, for more details, [23–25, 33, 35, 38]). In recent years, we have studied constacyclic codes of length ps over Fpm+uFpm. All constacyclic codes of length ps over the ring Fpm+uFpm are considered. The purpose of this subsection is to investigate the structure of all skew Θ-λ-constacyclic codes over Fpm+uFpm, where λ is fixed by Θ and the length n of codes is a multiple of the order of Θ. Note that the set of automorphisms of Fpm+uFpm forms a group under composition, denoted by Aut(Fpm+uFpm). The group Aut(Fpm+uFpm) is completely characterized by Alkhamees [37] as follows.Theorem 51.
Forα∈Aut(Fpm) and β∈Fpm∗, let (53)Θα,β:Fpm+uFpm⟶Fpm+uFpmbe defined by (54)Θα,βa+bu=αa+βαbu.
ThenAut(Fpm+uFpm)={Θα,β∣α∈Aut(Fpm),β∈Fpm∗}.In the next part, the structure of skewΘ-cyclic and negacyclic codes over Fpm+uFpm is studied. We refer the readers to [26, 38] for more details.Assume thatC is a nonzero left ideal in (Fpm+uFpm)[x;Θ]/〈xn-λ〉. Let A be the set of all nonzero skew polynomials of minimal degree in C. Then the classifications of Θ-λ-constacyclic codes are given in terms of generators of left ideals in (Fpm+uFpm)[x;Θ]/〈xn-λ〉.Theorem 52 (see [26, Theorem 4.1]).
LetC and A be as above. Then consider the following: (i)
If there exists a monic skew polynomial inA, then it is unique in A. In this case, C=〈g(x)〉, where g(x) is such unique skew polynomial.(ii)
If there are no monic skew polynomials inC, then there exists a unique skew polynomial g(x)=ug1(x) in A with leading coefficient u. In this case, C=〈g(x)〉.(iii)
If there are no monic skew polynomials inA but there exists a monic skew polynomial in C, then there exists a unique skew polynomial g(x)=ug1(x) in A with leading coefficient u and a unique monic skew polynomial f(x)=f0(x)+uf1(x) of minimal degree in C such that deg⁡(f1(x))<deg⁡(g1(x)). In this case, C=〈g(x),f(x)〉.We categorize the left ideals of(Fpm+uFpm)[x;Θ]/〈xn-λ〉 into three types: Type LI-1 refers to the trivial ideal (〈0〉,〈1〉) or a left ideal satisfying part (i) of the theorem above. Similarly, LI-2 and LI-3 refer to a left ideal satisfying Theorem 52 ((ii) and (iii)), respectively. Next, we provide some properties of left ideals of each type LI-i  (i=1,2,3.) First, we consider type LI-i by the following lemmas.Lemma 53 (see [26, Proposition 4.1]).
A left ideal of type LI-1 is principal and generated by a monic right divisorg(x) of xn-λ in (Fpm+uFpm)[x;Θ]. Moreover, if we view g(x)=g0(x)+ug1(x), where g0(x),g1(x)∈Fpm[x;Θ], then deg⁡(g1(x))<deg⁡(g0(x)) and g0(x) is a monic right divisor of xn-λ¯ in Fpm[x;Θ].Lemma 54 (see [26, Proposition 4.2]).
A left ideal of type LI-2 is principal and generated byg(x)=ug1(x), where g1(x) is a monic right divisor of xn-λ¯ in Fpm[x;Θ] such that deg⁡(g1(x))<n.We writef(x)← to indicate that f(x)← is the skew polynomial such that f(x)u=uf(x)←.Lemma 55 (see [26, Proposition 4.3]).
A left ideal of type LI-3 is generated by{g(x)=ug1(x),f(x)=f0(x)+uf1(x)}, where f0(x),f1(x),g1(x)∈Fpm[x;Θ] satisfy the following properties: (i)
g1(x), f0(x) are monic,(ii)
deg⁡(f1(x))<deg⁡(g1(x))<deg⁡(f0(x))<n,(iii)
g1(x) is a right divisor of f0(x) in Fpm[x;Θ],(iv)
f0(x) is a right divisor of xn-λ¯ in Fpm[x;Θ]. Moreover, if λ∈Fpm, then g1(x) is a right divisor of ((xn-λ)/f0(x))←f1(x) in Fpm[x;Θ].Example 56.
LetR=F3+uF3 be a finite chain ring. We consider the automorphism Θid,2 of F3+uF3, where Θid,2(a+ub)=a+2bu for all a,b∈F3. We list all left ideals in three types LI-i (i=1,2,3) in (F3+uF3)[x;Θid,2]/〈x2-1〉. All left ideals in type LI-1 are 〈0〉,〈1〉,〈x+1〉,〈x+2〉,〈x+1+u〉,〈x+1+2u〉,〈x+2+u〉,〈x+2+2u〉. All left ideals in type LI-2 are 〈u〉,〈u(x+1)〉,〈u(x+2)〉, and all left ideals in type LI-3 are 〈u,x+1〉,〈u,x+2〉.Applying Theorem52, the structure of skew Θ-λ-constacyclic codes over Fpm+uFpm is introduced. We have three types of the left ideals in the ring (Fpm+uFpm)[x;Θ]/〈xn-λ〉. From this, we study the structure of the Euclidean dual codes of skew Θ-cyclic and negacyclic codes over Fpm+uFpm.Theorem 57 (see [26, Theorem 4.2]).
Letλ∈{-1,1}. Then the Euclidean dual code of a left ideal in (Fpm+uFpm)[x;Θ]/〈xn-λ〉 is also a left ideal in (Fpm+uFpm)[x;Θ]/〈xn-λ〉 determined as follows:(LI-1⊥)
IfC=〈g0(x)+ug1(x)〉, then C⊥=〈xn-deg⁡(g0(x))φ((xn-λ)/(g0x+ug1x))〉.(LI-2⊥)
IfC=〈ug1(x)〉, then C⊥=〈u,xn-deg(g1(x))φ((xn-λ)/g1(x))〉.(LI-3⊥)
IfC=〈ug1(x),f0(x)+uf1(x)〉, then there exists m(x)∈Fpm[x;Θ] such that m(x)g1(x)=((xn-λ)/f0(x))←f1(x) and (55)C⊥=xn-deg⁡f0xφxn-λ/f0xu,xn-deg⁡g1xφxn-λg1x-umx,whereφ:R[x;Θ]→R[x;Θ]S-1 defined by φ(∑i=0k‍aixi)=∑i=0k‍x-iai.For Hermitian dual codes, we assume that the order ofΘ is 2. We have the structure of Hermitian dual codes of skew Θ-cyclic and negacyclic codes over Fpm+uFpm as follows.Theorem 58 (see [26, Theorem 4.3]).
Letλ∈{-1,1} and let Θ be an automorphism of order 2. Then the Hermitian dual code of a left ideal in (Fpm+uFpm)[x;Θ]/〈xn-λ〉 is also a left ideal in (Fpm+uFpm)[x;Θ]/〈xn-λ〉 determined as follows:(LI-1⊥)
IfC=〈g0(x)+ug1(x)〉, then C⊥H=〈ϕ(xn-deg(g0(x))φ((xn-λ)/(g0x+ug1x)))〉.(LI-2⊥)
IfC=〈ug1(x)〉, then C⊥H=〈u,ϕ(xn-deg(g1(x))φ((xn-λ)/g1(x)))〉.(LI-3⊥)
IfC=〈ug1(x),f0(x)+uf1(x)〉, then there exists m(x)∈Fpm[x;Θ] such that m(x)g1(x)=((xn-λ)/f0(x))←f1(x) and(56)C⊥=xn-deg⁡f0xφxn-λf0xu,xn-deg⁡g1xφxn-λg1x-u·mx,whereφ:Rx;Θ→Rx;ΘS-1 defined by φ∑i=0k‍aixi=∑i=0k‍x-iai.Finally, we give an example for Euclidean and Hermitian dual codes.Example 59.
We knew in previous example that all left ideals of type LI-1 in(F3+uF3)[x;Θid,2]/〈x2-1〉 are 〈0〉,〈1〉,〈x+1〉,〈x+2〉,〈x+1+u〉,〈x+2+u〉,〈x+1+2u〉,〈x+2+2u〉. Then their Euclidean dual codes are 〈1〉,〈0〉,〈x+2〉,〈x+1〉,〈x+2+u〉,〈x+1+u〉,〈x+2+2u〉,〈x+1+2u〉, respectively. Similarly, Hermitian dual codes are 〈1〉,〈0〉,〈x+2〉,〈x+1〉,〈x+2+2u〉,〈x+2+u〉,〈x+1+2u〉,〈x+1+u〉. All left ideals in type LI-2 are 〈u〉,〈u(x+1)〉,〈u(x+2)〉. The Euclidean dual codes coincided with the Hermitian dual codes of all left ideals in type LI-2. They are 〈u〉,〈u,x+2〉, and 〈u,x+1〉, respectively. Similarly, the Euclidean dual codes also coincided with the Hermitian dual codes of all left ideals in type LI-3. They are 〈u(x+1)〉,〈u(x+2)〉. We summarize discussion above in Table 3.Table 3
C
C
⊥
C
⊥
H
〈
0
〉
1
〈
1
〉
1
〈
1
〉
1
〈
1
〉
1
〈
0
〉
1
〈
0
〉
1
〈
x
+
1
〉
1
〈
x
+
2
〉
1
〈
x
+
2
〉
1
〈
x
+
2
〉
1
〈
x
+
1
〉
1
〈
x
+
1
〉
1
〈
x
+
1
+
u
〉
1
〈
x
+
2
+
u
〉
1
〈
x
+
2
+
2
u
〉
1
〈
x
+
1
+
2
u
〉
1
〈
x
+
2
+
2
u
〉
1
〈
x
+
2
+
u
〉
1
〈
x
+
2
+
u
〉
1
〈
x
+
1
+
u
〉
1
〈
x
+
1
+
2
u
〉
1
〈
x
+
2
+
2
u
〉
1
〈
x
+
1
+
2
u
〉
1
〈
x
+
1
+
u
〉
1
〈
u
〉
2
〈
u
〉
2
〈
u
〉
2
〈
u
(
x
+
1
)
〉
2
〈
u
(
x
+
2
)
〉
3
〈
u
(
x
+
2
)
〉
3
〈
u
(
x
+
2
)
〉
2
〈
u
(
x
+
1
)
〉
3
〈
u
(
x
+
1
)
〉
3
〈
u
,
x
+
1
〉
3
〈
u
(
x
+
2
)
〉
2
〈
u
(
x
+
2
)
〉
2
〈
u
,
x
+
2
〉
3
〈
u
(
x
+
1
)
〉
2
〈
u
(
x
+
1
)
〉
2
Note that the subscripts 1, 2, and 3 indicate the types of ideals LI-1, LI-2, and LI-3, respectively.


## 5.1. SkewΘ-λ-Constacyclic Codes over Finite Chain Rings

For a skew polynomialg(x) in R[x;Θ], then a left ideal generated by g(x), denoted by gx, is in general not a two-sided ideal. However, if g(x)=xth(x)(t∈N0) such that h(x) is central (i.e., commutes with all elements of R[x;Θ]), then 〈f(x)〉 is a principal two-sided ideal in R[x;Θ]. From this remark, the following corollary is a direct consequence.Corollary 34 (see [26, Corollary 2.2]).
Iff(x) is a monic central skew polynomial of degree n, then the skew polynomials of degree less than n are canonical representatives of the elements in R[x;Θ]/〈f(x)〉.Analogous to classical constacyclic codes, we study skewΘ-λ-constacyclic codes as left ideals in R[x;Θ]/xn-λ. Note that R[x;Θ] is a noncommutative ring. So we need to have the conditions of Θ and λ which ensure that xn-λ is a two-sided ideal.Lemma 35 (see [26, Proposition 2.2]).
Letn be a positive integer and λ a unit in R. Then the following statements are equivalent: (i)
xn-λ is central in R[x;Θ].(ii)
xn-λ is a two-sided ideal.(iii)
n is a multiple of the order of Θ and λ is fixed by Θ.ForΘ-λ-constacyclic codes over finite fields, a code C is a skew Θ-λ-constacyclic code if and only if C is a left ideal 〈g(x)〉⊆Fpm[x;Θ]/xn-λ, where g(x) is right divisor of xn-λ. In the case finite chain rings, the following theorem is analogous to that for Θ-λ-constacyclic codes over finite fields.Theorem 36 (see [26, Theorem 2.2]).
LetΘ be an automorphism of R, n an integer divisible by the order of Θ, and λ a unit in R which is fixed by Θ. Then the code C is a skew Θ-λ-constacyclic code if and only if C is a left ideal 〈g(x)〉⊆R[x;Θ]/〈xn-λ〉, where g(x) is a right divisor of xn-λ.From this theorem, we can find a skewΘ-λ-constacyclic code as a left ideal 〈g(x)〉⊆R[x;Θ]/xn-λ, where g(x) is a right divisor of xn-λ. However, it is not easy to list all skew Θ-λ-constacyclic codes because R[x;Θ] is not unique factorization ring. Therefore, there are many more right factors than in the commutative case, which in turn produces many more skew Θ-λ-constacyclic codes.Example 37.
LetR=F3+uF3 be a finite chain ring. We consider the automorphism Θid,2 of F3+uF3, where Θid,2(a+ub)=a+2bu. Then we have two irreducible factorizations of x6-1 in (F3+uF3)[x;Θid,2]: (45)x6-1=x+13x+23=x2+ux+23.Given a monic right divisor of degreen-k of xn-λ:g(x)=∑i=0n-k-1‍gi(x)+xn-k, then a generator matrix of the Θ-λ-constacyclic code C generated by g(x) is given by (46)G≔g0⋯gn-k-110⋯00Θg0⋯Θgn-k-11⋯00⋯⋯⋯Θ2gn-k-1⋯0⋮⋮⋮⋮⋮⋮⋮0⋯0Θk-1g0⋯Θk-1gn-k-11.The rows of G are linearly independent. Then we have the following result.Proposition 38 (see [26, Proposition 3.1]).
Letg(x) be a right divisor of xn-λ. Then the Θ-λ-constacyclic code C generated by g(x) is a free R-module with |C|=|R|n-deg⁡(g(x)).Similarly, in the case of finite fields, we denoteRΘ, the subring of R fixed by Θ. Then we have the following result.Proposition 39 (see [26, Proposition 3.2]).
Letg(x) be a monic right divisor of xn-λ in R[x;Θ]. The skew Θ-λ-constacyclic code generated by g(x) is λ-constacyclic if and only if g(x)∈RΘ[x;Θ].LetC be a Θ-λ-constacyclic code. In the following lemma, the parity-check matrix for C is introduced.Lemma 40 (see [26, Proposition 3.3]).
LetC be the Θ-λ-constacyclic code generated by a monic right divisor g(x) of xn-λ and h(x)≔(xn-λ)/g(x). Then the following statements hold: (i)
Forc(x)∈R[x;Θ], c(x)∈C if and only if c(x)h(x)=0 in R[x;Θ]/xn-λ.(ii)
Ifh(x)=∑i=0k-1‍hixi+xk, then the matrix (47)H≔1Θhk-1⋯Θkh00⋯001Θ2hk-1⋯Θk+1h0⋯000⋯⋯⋯⋯0⋮⋮⋮⋮⋮⋮⋮00⋯1Θn-khk-1⋯Θn-1h0is a parity-check matrix forC.In the next part, we study the Euclidean and Hermitian dual codes of skewΘ-λ-constacyclic codes over finite chain rings. Suppose that the length n of codes is divisible by the order of Θ, and λ is a unit in R which is fixed by Θ. Euclidean inner product is defined by u,v=∑i=0n-1‍uivi, for u=(u0,u1,…,un-1) and v=(v0,v1,…,vn-1) in Rn. In special case, if the order of Θ is 2, then we can also give the Hermitian inner product, denoted by u,vH=∑i=0n-1‍uiΘ(vi). If u,v=0 (resp., u,vH=0), then u and v are called Euclidean orthogonal (resp., Hermitian orthogonal). The Euclidean and Hermitian dual code of a code C are defined to be (48)C⊥=v∈Rn∣v,c=0,∀c∈C,(49)C⊥H=v∈Rn∣v,cH=0,∀c∈C,respectively. If C=C⊥ (C=C⊥H), then C is said to be Euclidean (Hermitian) self-dual code. We get a main result which describes the relationship between a skew Θ-λ-constacyclic code and its dual.Lemma 41 (see [26, Lemma 3.1]).
LetC be a code of length n over R. Then C is skew Θ-λ-constacyclic if and only if C⊥ is Θ-λ-1-constacyclic. In particular, if λ2=1, then C is Θ-λ-constacyclic if and only if C⊥ is Θ-λ-constacyclic.


## 5.2. Euclidean Dual Codes

We denote thatR[x;Θ]S-1 is the right localization of R[x;Θ]. The following theorem will discuss the necessary and sufficient conditions for R[x;Θ] to have the right localization.Theorem 42 (see [26, Theorem 2.1]).
LetS={xi∣i∈N}. Then R[x;Θ] has the right localization at S if and only if both the following conditions hold: (i)
For allxi∈S and a(x)∈R[x;Θ], there exist xj∈S and b(x)∈R[x;Θ] such that a(x)xi=xjb(x).(ii)
Givena(x)∈R[x;Θ] and xi∈S, if xia(x)=0, then there exists xj∈S such that a(x)xj=0.Before determining the structure of dual codes, we get the following result.Lemma 43 (see [26, Proposition 2.4]).
Letφ:R[x;Θ]→R[x;Θ]S-1 defined by (50)φ∑i=0kaixi=∑i=0kx-iai.Then φ is a ring antimonomorphism.From Lemma41, it is easy to verify that the Euclidean dual of a skew Θ-λ-constacyclic code C is again a skew Θ-λ-constacyclic code. We introduce a result about Euclidean dual codes. To do that, we need the following lemma.Lemma 44 (see [26, Lemma 3.2]).
Assume thatλ2=1. Let a(x)=a0+a1x+⋯+an-1xn-1 and b(x)=b0+b1x+⋯+bn-1xn-1 be polynomials in R[x;Θ]. Then the following statements are equivalent: (i)
The coefficient vector ofa(x) is Euclidean orthogonal to the coefficient vector of xi(xn-1φ(b(x))) for all i∈{0,1,…,n-1}, where φ:R[x;Θ]→R[x;Θ]S-1 is a ring antimonomorphism defined in Lemma 43.(ii)
(a0,a1,…,an-1) is Euclidean orthogonal to (bn-1,Θ(bn-2),…,Θn-1(b0)) and all its Θ-λ-constacyclic shifts.(iii)
a(x)b(x)=0 in R[x;Θ]/xn-λ.Theorem 45 (see [26, Theorem 3.3]).
Assume thatλ2=1. Let g(x) be a right divisor of xn-λ and h(x)≔(xn-λ)/g(x). Let C be the Θ-λ-constacyclic code generated by g(x). Then the following statements hold: (i)
The skew polynomialxdeg⁡(h(x))φ(h(x)) is a right divisor of xn-λ.(ii)
The Euclidean dualC⊥ is a Θ-λ-constacyclic code generated by xdeg⁡(h(x))φ(h(x)).Theorem 46 (see [26, Theorem 3.4]).
Assume thatλ2=1 and n is even, denoted by n=2k. Let g(x)=∑i=0k-1‍gixi+xk be a right divisor of xn-λ. Then the Θ-λ-constacyclic code generated by g(x) is Euclidean self-dual if and only if (51)∑i=0k-1gixi+xkΘ-kg0-1+∑i=0k-1Θi-kg0-1gk-ixi+xk=xn-λ.Theorem46 provided the necessary and sufficient conditions for a skew Θ-λ-constacyclic code to be Euclidean self-dual code. By applying Theorem 46, we can see that if the order of Θ divides k and λ≠-1, then there are no Euclidean self-dual skew constacyclic codes of length 2k. Moreover, if Θ is the identity automorphism and λ≠-1, then there are no Euclidean self-dual codes.


## 5.3. Hermitian Dual Codes

The Hermitian inner product is defined only when the order ofΘ is 2. Therefore, in this subsection, we always suppose that the order of Θ is 2. We first have some characterizations of Hermitian duality.Lemma 47 (see [26, Lemma 3.5]).
LetC be a code of length n over R. Then C is skew Θ-λ-constacyclic if and only if C⊥H is Θ-λ-1-constacyclic. In particular, if λ2=1, then C is Θ-λ-constacyclic if and only if C⊥H is Θ-λ-constacyclic.Letϕ be a ring automorphism of R[x;Θ] defined by ϕ(∑i=0s‍aixi)=∑i=0s‍Θ(ai)xi. Then we have the following result.Lemma 48 (see [26, Lemma 3.6]).
Assume thatλ2=1. Let a(x)=a0+a1x+⋯+an-1xn-1 and b(x)=b0+b1x+⋯+bn-1xn-1 be polynomials in R[x;Θ]. Then the following statements are equivalent: (i)
The coefficient vector ofa(x) is Euclidean orthogonal to the coefficient vector of xiϕ(xn-1φ(b(x))) for all i∈{0,1,…,n-1}, where φ:R[x;Θ]→R[x;Θ]S-1 is a ring antimonomorphism defined in Lemma 43.(ii)
(a0,a1,…,an-1) is Hermitian orthogonal to (Θ-1(bn-1),(bn-2),…,Θn-2(b0)) and all its Θ-λ-constacyclic shifts.(iii)
a(x)b(x)=0 in R[x;Θ]/〈xn-λ〉.Theorem 49 (see [26, Theorem 3.7]).
Assume thatλ2=1. Let g(x) be a right divisor of xn-λ and h(x)≔(xn-λ)/g(x). Let C be the Θ-λ-constacyclic code generated by g(x). Then the following statements hold: (i)
The skew polynomialϕ(xdeg(h(x))φ(h(x))) is a right divisor of xn-λ.(ii)
The Hermitian dualC⊥H is a Θ-λ-constacyclic code generated by ϕ(xdeg⁡(h(x))φ(h(x))).Similar to the case of the Euclidean self-dual code, we have the necessary and sufficient conditions for aΘ-λ-constacyclic code to be Hermitian self-dual.Theorem 50 (see [26, Theorem 3.8]).
Assume thatλ2=1 and n is even, denoted by n=2k. Let g(x)=∑i=0k-1‍gixi+xk be a right divisor of xn-λ. Then the Θ-λ-constacyclic code generated by g(x) is Hermitian self-dual if and only if (52)∑i=0k-1gixi+xkΘ-k-1g0-1+∑i=0k-1Θi-k-1g0-1gk-ixi+xk=xn-λ.From this theorem, ifk is odd and λ≠-1, then there are no Hermitian self-dual Θ-λ-constacyclic codes of length 2k.


## 5.4. Skew Constacyclic Codes overFpm+uFpm

The class of finite chain rings of the formFpm+uFpm has been used widely as alphabets in certain constacyclic codes. It has been studied by many researchers (see, for more details, [23–25, 33, 35, 38]). In recent years, we have studied constacyclic codes of length ps over Fpm+uFpm. All constacyclic codes of length ps over the ring Fpm+uFpm are considered. The purpose of this subsection is to investigate the structure of all skew Θ-λ-constacyclic codes over Fpm+uFpm, where λ is fixed by Θ and the length n of codes is a multiple of the order of Θ. Note that the set of automorphisms of Fpm+uFpm forms a group under composition, denoted by Aut(Fpm+uFpm). The group Aut(Fpm+uFpm) is completely characterized by Alkhamees [37] as follows.Theorem 51.
Forα∈Aut(Fpm) and β∈Fpm∗, let (53)Θα,β:Fpm+uFpm⟶Fpm+uFpmbe defined by (54)Θα,βa+bu=αa+βαbu.
ThenAut(Fpm+uFpm)={Θα,β∣α∈Aut(Fpm),β∈Fpm∗}.In the next part, the structure of skewΘ-cyclic and negacyclic codes over Fpm+uFpm is studied. We refer the readers to [26, 38] for more details.Assume thatC is a nonzero left ideal in (Fpm+uFpm)[x;Θ]/〈xn-λ〉. Let A be the set of all nonzero skew polynomials of minimal degree in C. Then the classifications of Θ-λ-constacyclic codes are given in terms of generators of left ideals in (Fpm+uFpm)[x;Θ]/〈xn-λ〉.Theorem 52 (see [26, Theorem 4.1]).
LetC and A be as above. Then consider the following: (i)
If there exists a monic skew polynomial inA, then it is unique in A. In this case, C=〈g(x)〉, where g(x) is such unique skew polynomial.(ii)
If there are no monic skew polynomials inC, then there exists a unique skew polynomial g(x)=ug1(x) in A with leading coefficient u. In this case, C=〈g(x)〉.(iii)
If there are no monic skew polynomials inA but there exists a monic skew polynomial in C, then there exists a unique skew polynomial g(x)=ug1(x) in A with leading coefficient u and a unique monic skew polynomial f(x)=f0(x)+uf1(x) of minimal degree in C such that deg⁡(f1(x))<deg⁡(g1(x)). In this case, C=〈g(x),f(x)〉.We categorize the left ideals of(Fpm+uFpm)[x;Θ]/〈xn-λ〉 into three types: Type LI-1 refers to the trivial ideal (〈0〉,〈1〉) or a left ideal satisfying part (i) of the theorem above. Similarly, LI-2 and LI-3 refer to a left ideal satisfying Theorem 52 ((ii) and (iii)), respectively. Next, we provide some properties of left ideals of each type LI-i  (i=1,2,3.) First, we consider type LI-i by the following lemmas.Lemma 53 (see [26, Proposition 4.1]).
A left ideal of type LI-1 is principal and generated by a monic right divisorg(x) of xn-λ in (Fpm+uFpm)[x;Θ]. Moreover, if we view g(x)=g0(x)+ug1(x), where g0(x),g1(x)∈Fpm[x;Θ], then deg⁡(g1(x))<deg⁡(g0(x)) and g0(x) is a monic right divisor of xn-λ¯ in Fpm[x;Θ].Lemma 54 (see [26, Proposition 4.2]).
A left ideal of type LI-2 is principal and generated byg(x)=ug1(x), where g1(x) is a monic right divisor of xn-λ¯ in Fpm[x;Θ] such that deg⁡(g1(x))<n.We writef(x)← to indicate that f(x)← is the skew polynomial such that f(x)u=uf(x)←.Lemma 55 (see [26, Proposition 4.3]).
A left ideal of type LI-3 is generated by{g(x)=ug1(x),f(x)=f0(x)+uf1(x)}, where f0(x),f1(x),g1(x)∈Fpm[x;Θ] satisfy the following properties: (i)
g1(x), f0(x) are monic,(ii)
deg⁡(f1(x))<deg⁡(g1(x))<deg⁡(f0(x))<n,(iii)
g1(x) is a right divisor of f0(x) in Fpm[x;Θ],(iv)
f0(x) is a right divisor of xn-λ¯ in Fpm[x;Θ]. Moreover, if λ∈Fpm, then g1(x) is a right divisor of ((xn-λ)/f0(x))←f1(x) in Fpm[x;Θ].Example 56.
LetR=F3+uF3 be a finite chain ring. We consider the automorphism Θid,2 of F3+uF3, where Θid,2(a+ub)=a+2bu for all a,b∈F3. We list all left ideals in three types LI-i (i=1,2,3) in (F3+uF3)[x;Θid,2]/〈x2-1〉. All left ideals in type LI-1 are 〈0〉,〈1〉,〈x+1〉,〈x+2〉,〈x+1+u〉,〈x+1+2u〉,〈x+2+u〉,〈x+2+2u〉. All left ideals in type LI-2 are 〈u〉,〈u(x+1)〉,〈u(x+2)〉, and all left ideals in type LI-3 are 〈u,x+1〉,〈u,x+2〉.Applying Theorem52, the structure of skew Θ-λ-constacyclic codes over Fpm+uFpm is introduced. We have three types of the left ideals in the ring (Fpm+uFpm)[x;Θ]/〈xn-λ〉. From this, we study the structure of the Euclidean dual codes of skew Θ-cyclic and negacyclic codes over Fpm+uFpm.Theorem 57 (see [26, Theorem 4.2]).
Letλ∈{-1,1}. Then the Euclidean dual code of a left ideal in (Fpm+uFpm)[x;Θ]/〈xn-λ〉 is also a left ideal in (Fpm+uFpm)[x;Θ]/〈xn-λ〉 determined as follows:(LI-1⊥)
IfC=〈g0(x)+ug1(x)〉, then C⊥=〈xn-deg⁡(g0(x))φ((xn-λ)/(g0x+ug1x))〉.(LI-2⊥)
IfC=〈ug1(x)〉, then C⊥=〈u,xn-deg(g1(x))φ((xn-λ)/g1(x))〉.(LI-3⊥)
IfC=〈ug1(x),f0(x)+uf1(x)〉, then there exists m(x)∈Fpm[x;Θ] such that m(x)g1(x)=((xn-λ)/f0(x))←f1(x) and (55)C⊥=xn-deg⁡f0xφxn-λ/f0xu,xn-deg⁡g1xφxn-λg1x-umx,whereφ:R[x;Θ]→R[x;Θ]S-1 defined by φ(∑i=0k‍aixi)=∑i=0k‍x-iai.For Hermitian dual codes, we assume that the order ofΘ is 2. We have the structure of Hermitian dual codes of skew Θ-cyclic and negacyclic codes over Fpm+uFpm as follows.Theorem 58 (see [26, Theorem 4.3]).
Letλ∈{-1,1} and let Θ be an automorphism of order 2. Then the Hermitian dual code of a left ideal in (Fpm+uFpm)[x;Θ]/〈xn-λ〉 is also a left ideal in (Fpm+uFpm)[x;Θ]/〈xn-λ〉 determined as follows:(LI-1⊥)
IfC=〈g0(x)+ug1(x)〉, then C⊥H=〈ϕ(xn-deg(g0(x))φ((xn-λ)/(g0x+ug1x)))〉.(LI-2⊥)
IfC=〈ug1(x)〉, then C⊥H=〈u,ϕ(xn-deg(g1(x))φ((xn-λ)/g1(x)))〉.(LI-3⊥)
IfC=〈ug1(x),f0(x)+uf1(x)〉, then there exists m(x)∈Fpm[x;Θ] such that m(x)g1(x)=((xn-λ)/f0(x))←f1(x) and(56)C⊥=xn-deg⁡f0xφxn-λf0xu,xn-deg⁡g1xφxn-λg1x-u·mx,whereφ:Rx;Θ→Rx;ΘS-1 defined by φ∑i=0k‍aixi=∑i=0k‍x-iai.Finally, we give an example for Euclidean and Hermitian dual codes.Example 59.
We knew in previous example that all left ideals of type LI-1 in(F3+uF3)[x;Θid,2]/〈x2-1〉 are 〈0〉,〈1〉,〈x+1〉,〈x+2〉,〈x+1+u〉,〈x+2+u〉,〈x+1+2u〉,〈x+2+2u〉. Then their Euclidean dual codes are 〈1〉,〈0〉,〈x+2〉,〈x+1〉,〈x+2+u〉,〈x+1+u〉,〈x+2+2u〉,〈x+1+2u〉, respectively. Similarly, Hermitian dual codes are 〈1〉,〈0〉,〈x+2〉,〈x+1〉,〈x+2+2u〉,〈x+2+u〉,〈x+1+2u〉,〈x+1+u〉. All left ideals in type LI-2 are 〈u〉,〈u(x+1)〉,〈u(x+2)〉. The Euclidean dual codes coincided with the Hermitian dual codes of all left ideals in type LI-2. They are 〈u〉,〈u,x+2〉, and 〈u,x+1〉, respectively. Similarly, the Euclidean dual codes also coincided with the Hermitian dual codes of all left ideals in type LI-3. They are 〈u(x+1)〉,〈u(x+2)〉. We summarize discussion above in Table 3.Table 3
C
C
⊥
C
⊥
H
〈
0
〉
1
〈
1
〉
1
〈
1
〉
1
〈
1
〉
1
〈
0
〉
1
〈
0
〉
1
〈
x
+
1
〉
1
〈
x
+
2
〉
1
〈
x
+
2
〉
1
〈
x
+
2
〉
1
〈
x
+
1
〉
1
〈
x
+
1
〉
1
〈
x
+
1
+
u
〉
1
〈
x
+
2
+
u
〉
1
〈
x
+
2
+
2
u
〉
1
〈
x
+
1
+
2
u
〉
1
〈
x
+
2
+
2
u
〉
1
〈
x
+
2
+
u
〉
1
〈
x
+
2
+
u
〉
1
〈
x
+
1
+
u
〉
1
〈
x
+
1
+
2
u
〉
1
〈
x
+
2
+
2
u
〉
1
〈
x
+
1
+
2
u
〉
1
〈
x
+
1
+
u
〉
1
〈
u
〉
2
〈
u
〉
2
〈
u
〉
2
〈
u
(
x
+
1
)
〉
2
〈
u
(
x
+
2
)
〉
3
〈
u
(
x
+
2
)
〉
3
〈
u
(
x
+
2
)
〉
2
〈
u
(
x
+
1
)
〉
3
〈
u
(
x
+
1
)
〉
3
〈
u
,
x
+
1
〉
3
〈
u
(
x
+
2
)
〉
2
〈
u
(
x
+
2
)
〉
2
〈
u
,
x
+
2
〉
3
〈
u
(
x
+
1
)
〉
2
〈
u
(
x
+
1
)
〉
2
Note that the subscripts 1, 2, and 3 indicate the types of ideals LI-1, LI-2, and LI-3, respectively.



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