Abstract
This paper overviews the study of skew --constacyclic codes over finite fields and finite commutative chain rings. The structure of skew --constacyclic codes and their duals are provided. Among other results, we also consider the Euclidean and Hermitian dual codes of skew -cyclic and skew -negacyclic codes over finite chain rings in general and over in particular. Moreover, general decoding procedure for decoding skew BCH codes with designed distance and an algorithm for decoding skew BCH codes are discussed.
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.
Shannon’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 to DETECT if there is an error and CORRECT 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. All -constacyclic codes of length are classified as ideals of , where is a divisor of . 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 is divisible by the characteristic 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 --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 are constructed and used for three applications: self-dual Euclidean codes give self-dual 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 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 .
The class of finite rings of the form has been widely used as alphabets of certain constacyclic codes. For example, the structure of is interesting; it is lying between and in the sense that it is additively analogous to and multiplicatively analogous to . 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 over the Galois extension rings of are classified and their detailed structures are also established in [25].
In 2012, Jitman et al. [26] introduced the notion of skew --constacyclic (or skew constacyclic) codes over finite chain rings. They studied the structure of skew --constacyclic, the Euclidean, and Hermitian dual codes of skew -cyclic and negacyclic codes over finite chain rings. The goal of this survey is to study skew --constacyclic codes over finite fields and finite chain rings.
This paper is arranged as follows. Basic concepts are reviewed in Section 2. After presenting preliminary concepts in Section 2, we study skew -negacyclic, cyclic, and --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 --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. Let be a finite field with multiplicative identity 1. The characteristic of is the least positive integer such that . Such always exists for a finite field and it is well known that the characteristic must be a prime.
Theorem 2. A finite field of characteristic contains elements for some integer . For every element of a finite field with elements, we have .
Definition 3. An element in a finite field is called a primitive element (or generator) of if .
Example 4. has primitive elements, namely, and . has primitive elements. In fact, expressing as , where , then and are primitive elements of .
Note that an automorphism of a field is a bijection such that and for all . Suppose that is a finite field of characteristic , and then the map is defined by , the Frobenius automorphism of . Since is a field of characteristic , we have . From , we can see that is a field homomorphism. Similarly, the map defined by is also a field homomorphism. The set of automorphisms of forms a group under composition which we denote as . Next, we give the following theorem characterizing this group.
Theorem 5 (see [27, Theorem ]). (i) If is a finite field, then is a cyclic group of order and is generated by Frobenius automorphism .
(ii) The prime subfield of is precisely the set of elements in such that .
(iii) The subfield of is precisely the set of elements in such that , where .
Theorem 6 (see [27, Theorem ]). If , then is a cyclic group of order and is generated by .
2.2. Codes, Cyclic Codes, Generator, and Parity-Check Matrices
Let be a finite field. A linear -code over is a -dimensional vector subspace of the vector space In this paper, all codes are assumed to be linear codes unless otherwise stated. We use polynomial representation of the code , where we identify codewords with coefficient tuples of polynomials: Those polynomials can also be seen as elements of a quotient ring , and any code of length over corresponds to a subset of .
Example 7. The polynomial of degree at most over finite field may be regarded as the word of length in . If , then the polynomial may be regarded as the word . Similarly, the polynomial may be regarded as the word .
Definition 8. Let be a word of length , and the cyclic shift the word of length : A code is said to be cyclic if , for all .
Example 9. Let be a linear code over . It is easy to see that , . This implies that is a linear cyclic code over .
Let be a linear code over . Since , we can conclude that is not cyclic code.
Definition 10. A code is said to be a -constacyclic code of length if it is closed under the -constacyclic shift defined by In particular, when or , such codes are called cyclic and negacyclic codes, respectively.
We now give some properties of cyclic code. The following results are well known (cf. [27]).
Theorem 11 (see [27, Theorem ]). Let be a nonzero cyclic code in . There exists a polynomial with the following properties:(i) is the unique monic polynomial of minimum degree in , and it is called the generating polynomial for .(ii).(iii)The generating polynomial divides .(iv)If , then has dimension and is a basis for .(v)Every element of is uniquely expressible as a product , where or , that is, .(vi)If , then and has the following generator matrix:
From this theorem, we can see that is a nonzero cyclic code in and is the monic polynomial of minimum degree in if and only if , and is divisible by .
Let be a cyclic code in with generator polynomial , such that . Let . Then a parity-check matrix for is given by
Example 12. Let be a cyclic code of length over the binary field . Put . Then we have . We can see that has dimension 3 and generating matrix is given by Hence, a parity-check matrix for is given by
Definition 13. (i) The Hamming distance between two vectors is defined to be the number of coordinates in which and differ.
(ii) The Hamming weight of a vector is the number of nonzero coordinates in .
(iii) For a code containing at least two words, the minimum distance of a code , denoted by , is
It is easy to see that the definition of distance satisfies nonnegativity, symmetry, and the triangle inequality, so our code is living in a metric space.
Example 14. Let be a binary code. Then we have We can see that This shows that .
The following theorem gives a relationship between minimum distance and the minimum weight of the nonzero codewords of a linear code .
Theorem 15 (see [27, Theorem ]). If , then . If is a linear code, the minimum distance is the same as the minimum weight of the nonzero codewords of .
2.3. The Skew Polynomial Ring
Now let be an automorphism of . We consider the set of formal polynomials where coefficients are written on the left of the variable . The set forms a ring under the usual addition of polynomials and the multiplication is defined by the following basic rule: . The multiplication is extended to all elements in by associativity and distributivity. The ring is called a skew polynomial ring over , and each element in is called a skew polynomial. It is easy to see that the ring is noncommutative unless is the identity automorphism on . If with , then we say that has degree , denoted by . The following facts are straightforward for the skew polynomial ring :(i)It has no nonzero zero-divisors,(ii)the units of are the units of ,(iii),(iv).Recall that a left (right) ideal of a ring is called a left (right) principal ideal if there exists an element such that , where . The element is called a generator of and is said to be generated by . A ring is called a left (right) principal ideal ring if every left (right) ideal of is principal. The skew polynomial ring is left and right Euclidean ring whose left and right ideals are principal. For which are nonzero, there exists unique polynomial such that . If , then is a right divisor of in . The definition of left divisor in is similar.
The centre of the skew polynomial ring is the set of all elements that commute with all other elements of . An element is called a central element. An automorphism is said to fix an element if . We denote the subfield of elements of which are fixed by . Then the ring is a commutative subring of . A polynomial is central element if and only if is both in and in , where . In other words, a polynomial is central element (i.e., commutes with all elements of ) if and only if , where is the order of [28, Theorem ].
3. Structure and Duals of Skew Constacyclic Codes over Finite Fields
In this section, we study skew --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 --constacyclic codes over finite fields.
Definition 16. Given an automorphism of and a unit in , a code is said to be skew --constacyclic of length if it is closed under the skew --constacyclic shift defined by
In particular, when or , such codes are called skew -cyclic and skew -negacyclic codes, respectively. When is the identity automorphism, they become classical constacyclic cyclic, cyclic, and negacyclic codes. A right factor of degree of generates linear code. While the ring is a commutative ring, so every ideal in is two-sided ideal, the skew polynomial ring is noncommutative. Therefore, we need to have conditions of and to ensure that is a two-sided ideal of . If is divisible by the order of and is fixed by , then is a two-sided ideal of . Indeed, for all in , we can see that Since is divisible by the order of , we have , . If is fixed by , then we have , proving that is in . This implies that is a two-sided ideal of , which makes the quotient ring well defined. If is not the identity, then 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 --constacyclic codes.
Example 17. Let be a generator of the multiplicative group of ; that is, is a zero of . Let be the automorphism of . We consider the polynomial . We have This shows that the ring is not a unique factorization ring.
Lemma in [3] can be extended as follows.
Lemma 18 (extending [3, Lemma ]). Let be an automorphism of an integer divisible by the order of , and a unit in which is fixed by . The ring is a principal left ideal ring, in which the left ideals are generated by , where is a right divisor of in .
Consider a codeword . ThenThus, is corresponding to a --constacyclic shift of , proving that the code is a skew --constacyclic code if and only if is a left ideal , where is a right divisor of . We summarize this discussion by the following theorem, which is an extension of [3, Theorem ].
Theorem 19 (extending [3, Theorem ]). Let be an automorphism of , an integer divisible by the order of , and a unit in which is fixed by . Then the code is a skew --constacyclic code if and only if is a left ideal , where is a right divisor of .
Given a monic right divisor of degree of , then a generator matrix of the --constacyclic code generated by is given by
Lemma 20 (see [29, Lemma ]). Let be an automorphism of , an integer divisible by the order of , and a unit in which is fixed by . Let be the --constacyclic code generated by a monic right divisor of and . If , then the following matrix is a parity-check matrix for .
Since for any , we have This shows that is fixed by . The following two corollaries are direct consequences of Theorem 19.
Corollary 21. Let be an automorphism of and an integer divisible by the order of . Then the code is a skew -negacyclic code if and only if is a left ideal , where is a right divisor of .
Corollary 22 (see [3, Lemma ]). Let be an automorphism of and an integer divisible by the order of . Then the code is a skew -cyclic code if and only if is a left ideal , where is a right divisor of .
We give an example to illustrate these results.
Example 23. Let be a generator of the multiplicative group of ; that is, is a zero of . Let be the automorphism of . To list all skew -cyclic codes over , we find all monic degree right factors of . They are Similarly, to list all skew -cyclic codes over , we find all monic degree right factors of . They are
Let be a right divisor of of degree . Then the skew -cyclic code is linear code with generator matrix
A right factor of degree of generates a linear code with parameters . If is not the identity, then the skew polynomial ring is in general not a unique factorization. In this case, we have more right factors than in the commutative case. For small values of , all right skew factors of 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 because many different codes with the same minimum distance can be found. A generating polynomial for one code respected the class of parameters is also exhibited. Table 1, computed by Bosma et al. [30], provides parameters and generating polynomials of skew -cyclic codes over , where is the Frobenius automorphism and is a generator of the multiplicative group of .
Given -tuples and , their inner product or dot product is defined in the usual way: evaluated in . Two codewords are called orthogonal if . For a linear code over , its dual code is the set of -tuples over that are orthogonal to all codewords of ; that is, A code is called self-orthogonal if , and it is called self-dual if . The following result is well known (cf. [29]).
Lemma 24 (see [29, Corollary ]). Let be an automorphism of , an integer divisible by the order of , and a unit in which is fixed by . Let and such that . The dual of the skew -cyclic code generated by in is the skew -cyclic code generated by
We give an example to illustrate how we use Lemma 24 to determine Euclidean self-dual -cyclic codes.
Example 25. Let be a generator of the multiplicative group of ; that is, is a zero of . Let be the automorphism of . We find all Euclidean self-dual -cyclic codes over in . From Example 23, we can list all monic degree 2 right factors of . Put to be all monic degree 2 right factors such that , . Then we have Applying Lemma 24, we have where the dual of the skew -cyclic code generated by in is the -cyclic code generated by . Suppose that , the skew -cyclic code generated by in , is an Euclidean self-dual -cyclic code. Then we have . This implies that is a constant multiple of . From this, the skew -cyclic codes generated by are Euclidean self-dual -cyclic codes.
We now turn our attention to Euclidean self-dual -cyclic codes over (cf. [29]). Suppose that is the Frobenius automorphism and is a generator of . It is easy to see that must be an even number. In fact, by Lemma 24, if is odd, then there are no Euclidean self-dual codes. Therefore, for some . Let be a self-dual code. Applying Lemma 24, the coefficients of the generating polynomial of is expressed. Since , and must differ by a constant multiple and . Now assume that with is the generator polynomial of the self-dual -cyclic code . Assume that such that . From Lemma 24, the code is generated by . Since is a constant multiple of . This implies that if the coefficients of both polynomials and are compared, then system (26) is built as follows:Since , it is easy to see that . By assumption, . This implies that . Hence, system (26) becomesSystem (27) allows expressing the coefficients of as follows: From , (28) becomes where powers of are of degree less than 4. By using the rule and expanding the skew product , polynomial equations in the coefficients of degree less than 4 in each variable can be determined. From , for any . Adding equations to polynomial equations in 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 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 -cyclic codes of length in will be exhibited. In [29], Boucher and Ulmer gave the table of Euclidean self-dual codes over and . We refer the readers to [29] for more details.
Recall that Hermitian inner product is denoted and calculated by , for all and in . Then we give the definition of the Hermitian dual code of a code as follows: If , then is said to be a Hermitian self-dual code. It is easy to check that if is odd, then there is no Hermitian self-dual of -cyclic codes. Therefore, must be an even number. Suppose that the order of divides . Let and be elements of such that . The Hermitian dual of a -cyclic code generated by in is again -cyclic code and is generated by [29, Lemma ]. Similar to the case of Euclidean self-dual codes, all the Hermitian self-dual -cyclic codes of length in can be found. The polynomial of Hermitian self-dual code in can be expressed as follows: In this case, the coefficient of (31) is shifted by . Expanding the skew product 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 and . For more details we refer the readers to [29].
4. Decoding Skew -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 . After that, we will modify the algorithm for decoding skew BCH codes.
Let be cyclic code over with generator polynomial of degree . Suppose that is transmitted and is received, where is the error vector with and . Let be the unique remainder when is divided by according to the Division Algorithm; that is, , where , with or . Then the function satisfies the following properties.
Theorem 26 (see [27, Theorem ]). With the preceding notation the following statements hold: (i) for all ; and all ,(ii),(iii) if and only if ,(iv)If , then ,(v)If , where and each have weight at most , then ,(vi) if .
Theorem 27 (see [27, Theorem ]). Let be a monic divisor of of degree . If , then , where is the coefficient of in .
Define the syndrome polynomial of any to be 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 polynomial of error patterns such that and .
Example 28 (see [27, Example 4.6.3]). Let be the binary cyclic code with a generating polynomial = , where is a 15th root of unity in . Then the syndrome polynomial of is . The syndrome polynomial for polynomial can be computed as follows. First, it is easy to see that . Then . Applying Theorem 26, = = = . Similarly, by applying Theorems 26 and 27, all syndrome polynomials will be determined. Table 2 shows all syndrome polynomials.
Step 2. Assume that is the received polynomial. Then the syndrome polynomial can be computed. Applying Theorem 26(iv), , where and .
Example 29 (see [27, Example 4.6.4]). We give an example for Step 2. We continue Example 28. Suppose that is a received polynomial. This implies that .
Step 3. If is in list computed in Step 1, then the error polynomial can be computed and it can be subtracted from to the corrected codeword . If 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 of until the syndrome polynomial is in the list from Step 1. If is in this list and it is associated with the error polynomial , then the received vector is decoded as . By using Theorem 27, and = = = .
We finish this part by the following example.
Example 30 (see [27, Example 4.6.6]). We can see that is not in the list computed in Step 1, then we continue to compute , which is not also appearing in the list in Step 1. It is easy to check that is in the list in Step 1. This implies that is decoded as .
Suppose that is a primitive th root of unity, is even, , and is an automorphism of such that . We give two results in [3] and use them later.
Lemma 31 (see [3, Proposition ]). For , and the remainder of the right division of by , then is a (classical) polynomial given by .
Lemma 32 (see [3, Proposition ]). Let be even, , and a primitive th root of unity. Let be a -cyclic code with . Let be its generating polynomial such that is a right divisor of in and is a right factor of for . The distance of the code is equal to its designed distance .
We now introduce the procedure for decoding skew BCH codes. Assume that is the error polynomial with , where . The polynomial is called a syndrome polynomial of . Note that is to be computed in the skew polynomial . Hence, where and . The polynomials and are called pseudolocator polynomial and evaluator polynomial, respectively. Let This implies that . This equation can be written to become , where .
Applying the Euclidean Algorithm to the polynomials and in , three sequences , and are defined as follows: and , , and with . The process will stop whenever can be determined satisfying and . From this, , , and can be computed by three equations as follows: From the roots of the pseudolocator polynomial , all , , will be listed. This shows that From the equation above, all coefficients of are also determined. For each , a finite number of possibilities solutions to the equation can be found. Similarly to the procedure for decoding BCH codes, this process will test until the skew polynomial is determined. Since 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]). Let be such that . Suppose that . Then the polynomial is a divisor of in . This implies that is the generator polynomial of a -cyclic code of length 10 over . We can see that is a right factor of for all . Hence, the designed distance of the code is . Now we consider and an error . The pertubed codeword is Since and polynomial , we have the syndrome polynomial Applying Euclid Algorithm to and in with , we can get the pseudolocator polynomial and the evaluator polynomial . The roots of the polynomial are , , and . From this, we have , , , and . By the polynomial , we can find , , and . Combining this result and the equations , we have , , and . Then we can list all possible errors as follows: It is easy to find that .
5. Skew --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 --constacyclic codes over finite chain rings. These codes have been studied for a particular case when codes are generated by monic right divisors of , where is a unit in the finite chain rings fixed by a given automorphism. Similarly to the case of skew --constacyclic codes over finite fields, when is the identity automorphism, they become classical constacyclic codes over finite chain rings. Therefore, skew --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 --constacyclic codes over finite chain rings is important. In this section, we overview the study of skew --constacyclic codes over finite chain rings studied by Jitman et al. [26].
A finite commutative ring with identity is called a finite 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 . Recently, skew -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 --constacyclic codes over finite fields and Galois rings to that over finite chain rings. The structure of all skew --constacyclic codes over a finite chain ring is determined. Moreover, Euclidean and Hermitian dual codes of skew -cyclic and negacyclic codes are considered. They also studied skew --constacyclic codes over a special case of a finite chain ring.
In this section, let be a finite chain ring with unique maximal ideal . Then is a nilpotent ideal of and we denote its nilpotency index by . Hence, the ideals of form the following chain:
Analogous to the case of finite fields, the set of automorphisms of forms a group under composition, denoted by . Many classes of finite chain rings have nontrivial automorphism groups. For examples, is nontrivial if and only if (cf. [16]) and is nontrivial if and only if or is odd or (cf. [37, Proposition ]).
We know that is left and right Euclidean ring whose left and right ideals are principal. Unlike the ring , if is a finite chain ring, then the skew polynomial ring is neither left nor right Euclidean ring. Therefore, we need to define left and right divisions. Suppose that and , where is a unit in and . We can see that the degree of polynomial is less than the degree of . By the inductive method, we can obtain skew polynomials and such that with or . If , then we say that is a right divisor of . The skew polynomials and are unique. They are called the right quotient and right remainder, respectively. Note that if , then we put . This algorithm is called the Right Division Algorithm in . The Left Division Algorithm in can be defined similarly, using the fact that the degree of is less than the degree of . Now we recall the definition of skew --constacyclic codes in . Given an automorphism of and a unit in , a linear code is said to be skew --constacyclic if is closed under the --constacyclic shift defined by
5.1. Skew --Constacyclic Codes over Finite Chain Rings
For a skew polynomial in , then a left ideal generated by , denoted by , is in general not a two-sided ideal. However, if such that is central (i.e., commutes with all elements of ), then is a principal two-sided ideal in . From this remark, the following corollary is a direct consequence.
Corollary 34 (see [26, Corollary ]). If is a monic central skew polynomial of degree , then the skew polynomials of degree less than are canonical representatives of the elements in .
Analogous to classical constacyclic codes, we study skew --constacyclic codes as left ideals in . Note that is a noncommutative ring. So we need to have the conditions of and which ensure that is a two-sided ideal.
Lemma 35 (see [26, Proposition ]). Let be a positive integer and a unit in . Then the following statements are equivalent: (i) is central in .(ii) is a two-sided ideal.(iii) is a multiple of the order of and is fixed by .
For --constacyclic codes over finite fields, a code is a skew --constacyclic code if and only if is a left ideal , where is right divisor of . In the case finite chain rings, the following theorem is analogous to that for --constacyclic codes over finite fields.
Theorem 36 (see [26, Theorem ]). Let be an automorphism of , an integer divisible by the order of , and a unit in which is fixed by Then the code is a skew --constacyclic code if and only if is a left ideal , where is a right divisor of .
From this theorem, we can find a skew --constacyclic code as a left ideal , where is a right divisor of . However, it is not easy to list all skew --constacyclic codes because is not unique factorization ring. Therefore, there are many more right factors than in the commutative case, which in turn produces many more skew --constacyclic codes.
Example 37. Let be a finite chain ring. We consider the automorphism of , where . Then we have two irreducible factorizations of in :
Given a monic right divisor of degree of , then a generator matrix of the --constacyclic code generated by is given by The rows of are linearly independent. Then we have the following result.
Proposition 38 (see [26, Proposition ]). Let be a right divisor of . Then the --constacyclic code generated by is a free -module with .
Similarly, in the case of finite fields, we denote , the subring of fixed by . Then we have the following result.
Proposition 39 (see [26, Proposition ]). Let be a monic right divisor of in . The skew --constacyclic code generated by is -constacyclic if and only if .
Let be a --constacyclic code. In the following lemma, the parity-check matrix for is introduced.
Lemma 40 (see [26, Proposition ]). Let be the --constacyclic code generated by a monic right divisor of and . Then the following statements hold: (i)For , if and only if in .(ii)If , then the matrix is a parity-check matrix for .
In the next part, we study the Euclidean and Hermitian dual codes of skew --constacyclic codes over finite chain rings. Suppose that the length of codes is divisible by the order of , and is a unit in which is fixed by . Euclidean inner product is defined by , for and in . In special case, if the order of is 2, then we can also give the Hermitian inner product, denoted by . If (resp., ), then and are called Euclidean orthogonal (resp., Hermitian orthogonal). The Euclidean and Hermitian dual code of a code are defined to be respectively. If (), then is said to be Euclidean (Hermitian) self-dual code. We get a main result which describes the relationship between a skew --constacyclic code and its dual.
Lemma 41 (see [26, Lemma ]). Let be a code of length over . Then is skew --constacyclic if and only if is --constacyclic. In particular, if , then is --constacyclic if and only if is --constacyclic.
5.2. Euclidean Dual Codes
We denote that is the right localization of . The following theorem will discuss the necessary and sufficient conditions for to have the right localization.
Theorem 42 (see [26, Theorem ]). Let . Then has the right localization at if and only if both the following conditions hold: (i)For all and , there exist and such that .(ii)Given and , if , then there exists such that .
Before determining the structure of dual codes, we get the following result.
Lemma 43 (see [26, Proposition ]). Let defined by Then is a ring antimonomorphism.
From Lemma 41, it is easy to verify that the Euclidean dual of a skew --constacyclic code is again a skew --constacyclic code. We introduce a result about Euclidean dual codes. To do that, we need the following lemma.
Lemma 44 (see [26, Lemma ]). Assume that . Let and be polynomials in . Then the following statements are equivalent: (i)The coefficient vector of is Euclidean orthogonal to the coefficient vector of for all , where is a ring antimonomorphism defined in Lemma 43.(ii) is Euclidean orthogonal to and all its --constacyclic shifts.(iii) in .
Theorem 45 (see [26, Theorem ]). Assume that . Let be a right divisor of and . Let be the --constacyclic code generated by . Then the following statements hold: (i)The skew polynomial is a right divisor of .(ii)The Euclidean dual is a --constacyclic code generated by .
Theorem 46 (see [26, Theorem ]). Assume that and is even, denoted by . Let be a right divisor of . Then the --constacyclic code generated by is Euclidean self-dual if and only if
Theorem 46 provided the necessary and sufficient conditions for a skew --constacyclic code to be Euclidean self-dual code. By applying Theorem 46, we can see that if the order of divides and , then there are no Euclidean self-dual skew constacyclic codes of length . Moreover, if is the identity automorphism and , then there are no Euclidean self-dual codes.
5.3. Hermitian Dual Codes
The Hermitian inner product is defined only when the order of is 2. Therefore, in this subsection, we always suppose that the order of is 2. We first have some characterizations of Hermitian duality.
Lemma 47 (see [26, Lemma ]). Let be a code of length over . Then is skew --constacyclic if and only if is --constacyclic. In particular, if , then is --constacyclic if and only if is --constacyclic.
Let be a ring automorphism of defined by . Then we have the following result.
Lemma 48 (see [26, Lemma ]). Assume that . Let and be polynomials in . Then the following statements are equivalent: (i)The coefficient vector of is Euclidean orthogonal to the coefficient vector of for all , where is a ring antimonomorphism defined in Lemma 43.(ii) is Hermitian orthogonal to and all its --constacyclic shifts.(iii) in .
Theorem 49 (see [26, Theorem ]). Assume that . Let be a right divisor of and Let be the --constacyclic code generated by . Then the following statements hold: (i)The skew polynomial is a right divisor of .(ii)The Hermitian dual is a --constacyclic code generated by .
Similar to the case of the Euclidean self-dual code, we have the necessary and sufficient conditions for a --constacyclic code to be Hermitian self-dual.
Theorem 50 (see [26, Theorem ]). Assume that and is even, denoted by . Let be a right divisor of . Then the --constacyclic code generated by is Hermitian self-dual if and only if
From this theorem, if is odd and , then there are no Hermitian self-dual --constacyclic codes of length .
5.4. Skew Constacyclic Codes over
The class of finite chain rings of the form 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 over . All constacyclic codes of length over the ring are considered. The purpose of this subsection is to investigate the structure of all skew --constacyclic codes over , where is fixed by and the length of codes is a multiple of the order of . Note that the set of automorphisms of forms a group under composition, denoted by . The group is completely characterized by Alkhamees [37] as follows.
Theorem 51. For and , let be defined by Then .
In the next part, the structure of skew -cyclic and negacyclic codes over is studied. We refer the readers to [26, 38] for more details.
Assume that is a nonzero left ideal in . Let be the set of all nonzero skew polynomials of minimal degree in . Then the classifications of --constacyclic codes are given in terms of generators of left ideals in .
Theorem 52 (see [26, Theorem ]). Let and be as above. Then consider the following: (i)If there exists a monic skew polynomial in , then it is unique in . In this case, , where is such unique skew polynomial.(ii)If there are no monic skew polynomials in , then there exists a unique skew polynomial in with leading coefficient . In this case, .(iii)If there are no monic skew polynomials in but there exists a monic skew polynomial in , then there exists a unique skew polynomial in with leading coefficient and a unique monic skew polynomial of minimal degree in such that . In this case, .
We categorize the left ideals of into three types: Type LI-1 refers to the trivial ideal 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- (.) First, we consider type LI- by the following lemmas.
Lemma 53 (see [26, Proposition ]). A left ideal of type LI-1 is principal and generated by a monic right divisor of in . Moreover, if we view , where , then and is a monic right divisor of in .
Lemma 54 (see [26, Proposition ]). A left ideal of type LI-2 is principal and generated by , where is a monic right divisor of in such that .
We write to indicate that is the skew polynomial such that .
Lemma 55 (see [26, Proposition ]). A left ideal of type LI-3 is generated by , where satisfy the following properties: (i), are monic,(ii),(iii) is a right divisor of in ,(iv) is a right divisor of in . Moreover, if , then is a right divisor of in .
Example 56. Let be a finite chain ring. We consider the automorphism of , where for all . We list all left ideals in three types LI- () in . All left ideals in type LI-1 are . All left ideals in type LI-2 are , and all left ideals in type LI-3 are .
Applying Theorem 52, the structure of skew --constacyclic codes over is introduced. We have three types of the left ideals in the ring . From this, we study the structure of the Euclidean dual codes of skew -cyclic and negacyclic codes over .
Theorem 57 (see [26, Theorem ]). Let . Then the Euclidean dual code of a left ideal in is also a left ideal in determined as follows:(LI-)If , then .(LI-)If , then .(LI-)If , then there exists such that and where defined by .
For Hermitian dual codes, we assume that the order of is 2. We have the structure of Hermitian dual codes of skew -cyclic and negacyclic codes over as follows.
Theorem 58 (see [26, Theorem ]). Let and let be an automorphism of order 2. Then the Hermitian dual code of a left ideal in is also a left ideal in determined as follows:(LI-)If , then .(LI-)If , then .(LI-)If , then there exists such that and where defined by .
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 are . Then their Euclidean dual codes are , respectively. Similarly, Hermitian dual codes are All left ideals in type LI-2 are . The Euclidean dual codes coincided with the Hermitian dual codes of all left ideals in type LI-2. They are , and , respectively. Similarly, the Euclidean dual codes also coincided with the Hermitian dual codes of all left ideals in type LI-3. They are . We summarize discussion above in Table 3.
Disclosure
The main part of this paper was written during the visits of Bac T. Nguyen to Hai Q. Dinh at Department of Mathematical Sciences, Kent State University, USA, from November 2014 to January 2015, and Hai Q. Dinh to Songsak Sriboonchitta at Faculty of Economics, Chiang Mai University, Thailand, in January 2015.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful for the Department of Mathematical Sciences, Kent State University; Faculty of Economics, Chiang Mai University; and Department of Mathematics, Mahidol University, for their hospitality and financial support. The authors’ thanks are extended to the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, for partial financial support. Bac T. Nguyen was supported by Postgraduate Student Exchange Scholarship, Mahidol University, to visit Department of Mathematical Sciences, Kent State University, USA.
Endnotes
- Claude Elwood Shannon (April 30, 1916–February 24, 2001) was an American mathematician, electronic engineer, and cryptographer, who is refered to as “the father of information theory” [39]. Shannon is also known as the founder of both digital computer and digital circuit design theory, when, as a 21-year-old M.S. student at MIT in 1937, he wrote a thesis establishing that electrical application of Boolean algebra could construct and resolve any logical, numerical relationship [40]. It has been claimed that this was the most important M.S. thesis of all time. Shannon contributed to the field of cryptanalysis during World War II and afterwards, including basic work on code breaking.
- Turbo codes were first introduced and developed in 1993 by Berrou et al. [41]. Turbo codes are a class of high-performance forward error correction codes, which were the first practical codes to closely approach the channel capacity, a theoretical maximum for the code rate at which reliable communication is still possible given a specific noise level. Turbo codes are widely used in deep space communications and other applications where designers seek to achieve reliable information transfer over bandwidth-constrained or latency-constrained communication links in the presence of data-corrupting noise. The first class of turbo code was the parallel concatenated convolutional code. Since the introduction of the original parallel turbo codes in 1993, many other classes of turbo code have been discovered, including serial versions and repeat-accumulate codes. Iterative turbo decoding methods have also been applied to more conventional forward error correction systems, including Reed-Solomon corrected convolutional codes.
- LDPC (low-density parity-check) codes were first introduced in 1963 by Gallager in his doctoral dissertation at MIT [42]. At that time, it was impractical to implement and LDPC codes were forgotten, but they were rediscovered in 1996. LDPC code is a linear error correcting code, a method of transmitting a message over a noisy transmission channel, and is constructed using a sparse bipartite graph. LDPC codes are capacity-approaching codes, which means that practical constructions exist that allow the noise threshold to be set arbitrarily close on the binary erasure channel to the Shannon limit for a symmetric memoryless channel. The noise threshold defines an upper bound for the channel noise, up to which the probability of lost information can be made as small as desired. Using iterative belief propagation techniques, LDPC codes can be decoded in time linear to their block length.
- Richard Wesley Hamming (February 11, 1915–January 7, 1998) was an American mathematician whose work had many implications for computer science and telecommunications. His contributions include the Hamming code (which makes use of a Hamming matrix), the Hamming window, Hamming numbers, sphere-packing (or Hamming bound), and the Hamming distance.
- During the late 1940s at Bell laboratories, Richard Hamming decided that a better system was needed. As folklore has it, Richard Hamming was working for Bell Labs. He was allowed to use the computer for research over the weekends. He would put together his punch cards during the week and submit them to be run over the weekend. This would work great as long as his punch cards were completely error-free; however, a single error would cause the computer to pass the job over and move on to the next. He would have to make corrections and resubmit his program at a later time. Richard Hamming thought that if the computer was smart enough to know that there was a mistake, why not have the computer find the mistake, correct it, and continue running the program. He then created the first error correction code, the Hamming Code. This not only solved an important problem in telecommunications and computer science, it opened up a whole new field of study.