Abstract
In the last few decades, the algebraic coding theory found widespread applications in various disciplines due to its rich fascinating mathematical structure. Linear codes, the basic codes in coding theory, are significant in data transmission. In this article, the authors’ aim is to enlighten the reader about the role of linear codes in a fuzzy environment. Thus, the reader will be aware of linear codes over lattice valued intuitionistic fuzzy type-3 (LIF-3) R-submodule and -intuitionistic fuzzy (-IF) submodule. The proof that the level set of LIF-3 is contained in the level set of -IF is given, and it is exclusively employed to define linear codes over -IF submodule. Further, -IF cyclic codes are presented along with their fundamental properties. Finally, an application based on genetic code is presented, and it is found that the technique of defining codes over -IF submodule is entirely applicable in this scenario. More specifically, a mapping from the module to a lattice (comprising 64 codons) is considered, and -IF codes are defined along with the respective degrees.
1. Introduction
The process of data hauling in the recent decades has been significantly escalated. To widen the utility of communication networks, the employment of intricate technologies such as wavelength division multiplexing has been enforced through the development in signal processing. Due to these advancements, a dire obstacle of unreliability in data transmission is faced; this may be impacted by the means of transmission or any other reason. For the maximum efficacy, there is an urge to control and process the glitches with the assurance of data transmission. To circumvent this problem, one may utilize the understanding of the rules to digitally interpret and store information. The system of rules for digital data transmission is known as a code. The algorithm that represents a sequence of numbers to detect and correct an error is called an error-correcting code. Initially, codes are defined over finite field , among which the binary codes are the simplest. Say, in a binary field , two codewords 110 and 111 are considered; then, both codewords comprise three bits. Now, when a message carrying 111 is sent in binary coding, the received message should contain 111 theoretically. Often, the transmission is corrupted, and the received message could be carrying 110. This error can be corrected by understanding the coding theory, and data transmission can be made more reliable and accurate. Linear codes are the most basic error-correcting codes which are the subspaces of a vector space. These codes are crucial to data transmission and data storage [1]. Just like vector spaces, codes are defined over groups and applied in channel and source coding [2–4].
Now, when dealing with the data transmission, one cannot always get crisp zeros and ones with data transmission ambiguity being an often occurring scenario. There may be plenty of reasons behind the emergence of uncertainty. These may be lack of knowledge, chance, imprecision, lack of information, and complexity. The imprecision inherent in the process can be dealt with by using fuzzy logic. Fuzzy sets (IF) are opposite to ordinary sets (crisp sets). In crisp set theory, things are categorized by the values 0 and 1, but the full membership and nonmembership are determined in a fuzzy set. This technique copes mathematically with the vagueness of determining boundaries by assigning grades of membership to the elements. Fuzzy set theory as a generalization of the classical set theory was established by Zadeh [5] in which each element under consideration is graded with a membership value ranging between zero and one. For instance, a membership extent shows that belongs to set with the degree 0.7 on a scale whereas zero indicates that there is no membership and one suggests complete membership. He also defined various properties of a fuzzy set such as union, intersection, and complements. This theory of fuzzy set was proved to be more effective against ambiguity. Atanassov [6] proposed the notion of an intuitionistic fuzzy set (IFS) which is a generalized concept of the fuzzy set governed by incorporating the degree of nonmembership along with the degree of membership. Goguen [7] discussed order structure and L-fuzzy set which is the generalization of fuzzy subset , as a function from subset to a lattice . Atanassov and Stoeva [8] proposed the definition of lattice valued intuitionistic fuzzy set (LIFS-1) where a complete lattice with a unary operation is considered. Gerstenkorn and Tepa cev [9] extended the concept formulated by Atanassov and defined LIFS-2 by replacing unary operation with unary operator by a linearization function . The choice of linearization creates a problem; to overcome this drawback, it is replaced by Lattice homomorphism , and it is said to be LIFS-3.
The Fuzzy set is the most suitable framework to model uncertain data which plays an important role in the data transmission. The vagueness in data transmission can be handled by involving fuzzy theoretic concepts in coding structures. Many researchers have worked on and proposed significant results related to this field. Kaenel and Pierre [10] considered n-dimensional vector space and defined fuzzy codes which are fuzzy subsets of n-tuples over the field. Hamming distance is also defined between two fuzzy codewords. Hall and Dial [11] investigated whether the distance between the fuzzy codeword and fuzzy subsets of n-tuples depends on the dimension of the space and distance between codewords which are non-fuzzy.
e elja and Tepav evi [12] introduced another method of involving fuzzy theory in coding based on defining a map from a non-empty set to partially ordered set . e elja et al. [13] used the concept and defined binary block codes over lattice valued fuzzy sets (-fuzzy sets). i ovi and Lazarevi [14] discussed the length and cardinality of block codes over -fuzzy sets. Amudhambigai and Neeraja [15] examined fuzzy codes and defined some basic operations including fuzzy complement, fuzzy intersection, and fuzzy union of fuzzy codes. Tsafack et al. [16] considered the Galois ring. They presented fuzzy linear codes and fuzzy cyclic codes. Shijina [17] investigated the notion of multi-fuzzy code, which is defined as multi-fuzzy subset of n-tuples over , and proposed fundamental properties of these codes. Hamming distance of multi-fuzzy codes was also presented.
The algebraic codes are also studied in various other disciplines and have a wide range of applications in numerous fields like data compression, cryptography, network processing, and neuroscience. Considerable work relevant to these fields has been done. Timm and Lapish [18] studied the encoding of information in neuroscience. To understand brain functions, it is important to know how a neural system integrates, encodes, and computes information. Various models were also analyzed to illustrate the strengths of the information theory analysis. Dong and Li [19] considered the linear network coding based qualitative communication and proved its importance. Kong et al. [20] discussed Alamouti code based on block repetition in FBMC/OQAM systems. A novel block-wise Alamouti code, where a repeated block is designed to remove the imaginary interference among FBMC/OQAM symbols, was presented. Marani [21] examined -invariant codes from primitive permutation representations of Mathieu groups M24 and M23.
In this article, LIFS-3 codes over R-submodule are defined. Module is a useful algebraic structure introduced as an extensions of vector space where scalars are from the arbitrary ring instead of a field. Fuzzy logic has significant importance in the theory of modules and rings. Remarkable work has been done by the researchers in this field. Further, Negoita and Ralescu [22] presented the concept of fuzzy modules. Many other researchers have also worked in this field and studied fuzzy modules. Zahedi [23] studied -fuzzy modules. He also presented basic operations on L-fuzzy modules such as addition and intersection and proved some properties related to these operations. Sharma [24] introduced the concept of an intuitionistic fuzzy module over an intuitionistic fuzzy ring. Sharma and Kanchan [25] in the continuation of this research work on the intuitionistic -fuzzy modules discussed some important results related to the L-fuzzy module and L-prime fuzzy module.
This article is based on the construction of linear codes over lattice valued intuitionistic fuzzy type-3 -submodule. Already existing fuzzy linear codes are defined over R-submodule which involves only the degree of membership. Although fuzzy set theory provides a convenient way to model uncertain data, in some situations, these are not more helpful when we need some extra information along with a membership degree. Lattice valued intuitionistic fuzzy sets which are the extension of fuzzy sets provide an effective tool to study the case of vagueness and have a significant role in various branches of mathematics such as group theory and module theory. Linear codes defined over lattice valued intuitionistic fuzzy type-3 R-submodule are more efficient as compared to ordinary fuzzy linear code over submodule because these involve both the degrees of belongingness and non-belongingness.
2. Preliminaries
In this section, some basic definitions will be discussed.
2.1. Linear Codes
Let be a finite field; then, is an -dimensional vector space over . A code over is simply a subset of . The elements of code are called the codewords, and these codewords are represented as . A code is said to be linear code over field if ; then, for all . Thus, is a subspace of . If under addition and multiplication modulo 2, then the code is said to be a binary code, and if the code is linear, then it is called a binary linear code. Let be a finite field. A code is called cyclic if the cyclic shift of each codeword in is also a member in . Let be a code over . Then, the corresponding dual code is defined to be
Thus, the dual code consists of all the codewords that are orthogonal to every codeword in .
2.2. Lattice Order Set
A relation on a non-empty set is is a subset of . If belongs to a relation , this implies is related to , or we can write it as . A relation on a set is said to be a partial order relation if it satisfies three properties, that is, reflexive property , antisymmetric property , and transitive property .
Consider two non-empty sets and , where is a subset of . An element is said to be the upper bound of provided that
In addition, the element is said to be lower bound of set if
If is the set of all upper bounds, then the least element of this set is called a supremum or join and if is the set of all lower bounds, then the greatest element of set is called an infimum or meet .
A set together with partial order is said to be a lattice if every set of two elements have supremum and infimum.
2.3. Lattice Valued Intuitionistic Fuzzy Set
The notion of fuzzy set introduced by Zadeh [1] is an extension of an ordinary set. Given universe , a fuzzy set is an ordered set (element of universe, degree of that element). Mathematically,
The grade of membership indicates the confirmation of an element that belongs to that set, but it does not give any information about the element which does not belong to that set. For this purpose, a generalization of the fuzzy set was introduced. For a given universe , an intuitionistic fuzzy set is a triplet that consists of an element of universe, value of membership of that element, and value of nonmembership of that element. Mathematically, it can be represented as
with .
Let be a non-empty set and be a lattice. Consider , , where and are membership and nonmembership functions; then, a lattice valued intuitionistic fuzzy set of type-1 (LIFS-1) is the set , where is an involutive order reversing unary operator on such that .
Let be a non-empty set and be a lattice. Consider , ; then, a lattice valued intuitionistic fuzzy set of type-2 (LIFS-2) is the set , where is a linearization function satisfying .
Let be a non-empty set and be a complete lattice with top element and bottom element . Consider , which are membership and nonmembership functions; then, a lattice valued intuitionistic fuzzy set of type-3 (LIFS-3) is the set , where is a lattice homomorphism with , satisfying
and for all .
2.4. -Module
Ring theory is one of the extensions of group theory that encompasses a broad set of present study topics in mathematics, computer science, and mathematical/theoretical physics. They have a wide range of applications in the studies of geometric objects and topology, and their connections to other fields of algebra are quite well understood in several contexts. A ring is a set equipped with two binary operations, namely, addition and multiplication and satisfying the following axioms:(1) is an Abelian group.(2) is a semigroup.(3)Multiplication is distributive over addition; that is, and for all .
The ring is called commutative if for all . Consider a commutative ring . An -module is a set together with a binary operation addition ‘’ and scalar multiplication ‘’, where ; then, for all , , we have the following:(1).(2).(3).(4) where is the multiplication identity of the ring .
Let be a ring; then, is an -module.
3. Lattice Valued Intuitionistic Fuzzy Type-3 -Submodule
If is a lattice valued intuitionistic fuzzy subset of type-3, where and , then is called an LIF-3 submodule of if for and we have the following:(1). (2).(3). (4).
Let be two composition functions, where and . Then, is called an -intuitionistic fuzzy (-IF) submodule if for and we have the following:(1).(2).(3).(4).
Remark 1. Ifis an LIF-3 submodule, thenMoreover, if is an -IF submodule, then
Proposition 1. Letbe an LIF-3 submodule. Then, the necessary and sufficient conditions forto be an-module are as follows:(1),(2), and .
Definition 1. An LIF-3 submoduleof a ringis called an LIF-3 ideal if for each,(1).(2).(3).(4).
Definition 2. Letbe an LIF-3 submodule; then, the level sets are defined asIn a similar fashion, we can define level sets for an -IF submodule as follows:We can also write that
3.1. LIF-3 Codes over Modules
If we consider a module which is a -module, then an LIF-3 submodule of is termed as an LIF-3 linear code having length over .
Proposition 2. Letbe an LIF-3 submodule andbe an-IF submodule. Letandbe the level sets forand, whereand. Then,.
Proof. Let and be two level sets for and . Let ; then, and imply that and , where . From this, we get that . Thus, . Let and ; then, .
From the above proposition, we have concluded that the level set of LIF-3 submodule is contained in -IF submodule, so we will use -IF submodule for further discussion of codes. Let us consider a module which is a module, and let be an -IF submodule; then, is said to be an -IF linear code of length over the module .
Example 1. Consider-module and latticehaving 0, 1 as bottom and top elements with, and. Letandbe defined as
Let , , and . Then, is an -IF module; therefore, is an -IF linear code.
Definition 3. Consider an -IF submodule ; then, the number of the elements mapped in to the same element, say, , is said to be the degree of that element and it is denoted by . In example 1, , , .
Proposition 3. Consider an-IF submodule; is an-IF linear code overiffare linear codes.
Definition 4. Let be an -IF submodule and be non-empty subset of . The -IF characteristic function of is denoted by , where ; then,
Theorem 1. Letbe a subset of an-IF submodule; then,is an-IF linear code overiffforis an-IF linear code over the same ring.
Proof. Let be a linear code implying that is an -IF submodule. Now, if , then by Definition 4 of , we haveAs is an -IF submodule, then and for all imply that and . Similarly, and . Thus, the of is an -IF submodule; hence, it is an -IF linear code.
Conversely, suppose that the be a linear code; hence, it is an -IF submodule. Let ; then, by Definition 4,imply that and . Similarly, and ; hence, ; thus, is an -IF submodule, so is an -IF linear code.
Proposition 4. Letbe an-IFS of. is an-IF linear code overiffis an-IF linear code.
Definition 5. Letbe an-IF fuzzy submodule; then,is also a linear code such that for, and; then,is said to be a trivial-IF linear code.
Definition 6. Letbe a module over the ring. Then, two-IF submodulesandare said to be orthogonal ifFurthermore, for all , we havewhere is the inner product on module .
Example 2. Letbe two-IF submodules which are defined asandAs, we can compute the valuesfor.Similarly, the remaining values given in Table 1 and2 can be obtained from the definition of level set.
Hence, is orthogonal to . We have shown the orthogonality of two sets, but under some conditions, the two sets are not orthogonal. This can be verified through the following remark.
Remark 2. Letbemodule and, be two-IF submodules. If for all,or, where, then setis not orthogonal to set.
Remark 3. Letbe an-IF submodule, sois a linear code overhaving length. Consider setsAs the set is finite, and are also finite. Consider an order ; then, the set satisfies this order. Similarly, suppose and this order is satisfied by . is a level set which is a linear code, where the generator matrix for this linear code is given by . Hence, can be obtained from the matrices .
Theorem 2. Letbe a finite module andbe an-IF submodule of. Then, there is an-IF submodulesuch thatis orthogonal toif and only if, and for any, there exists an elementwith. Similarly, for any element, there existsuch that.
Proof. Let be a module and be an -IF submodule of . Suppose that and and for any element there is an element such that . Moreover, for any element , there exists an element such that .Now, consider an order for and asBy using these compositions, we can also define sets which form partition of module aswhere . Now, if we define an -IF set , where , then for , we have and .
As we know, imply that . Similarly, for , we have . As for any element , there is an element such that . Similarly, for any , there is an element such that , as in finite module there is a property related to orthogonality by which . Accordingly, we have .
Furthermore, and . Thus, is an -IF submodule.
Conversely, suppose that and be two -IF submodules such that these two sets are orthogonal to each other; then, we have and . For all , we also have and ; then, for any element , there exist such that , and for any element , there exist such that ; this is due to the reason that and .
Theorem 3. Suppose that, , andbe three-IF fuzzy submodules of a modulesuch thatis orthogonal toandis orthogonal to; then,.
Proof. Let , , and be three -IF fuzzy submodules of a module such that is orthogonal to and is orthogonal to . Let , for . Then, for , . is orthogonal to , which implies that . Thus, . Therefore, . By following the same strategy, we get . Thus, . In a similar manner, we can show that = . Hence, .
Corollary 1. Letbe a finitemodule andbe an-IF submodule of; if there exists-IFSsuch thatis orthogonal to the submodule, thenis an-IF submodule of module.
Definition 7. Consider two-IF linear codesand; then, these two-IF codes are said to be equivalent if the level setsand(which are also linear codes) forandare equivalent.
3.2. -IF Cyclic Codes
Cyclic codes have long since been one of the most interesting families of codes because of their rich algebraic structure, and these codes play an important role in coding theory. In this section, we will discuss -IF cyclic codes and some important results related to these codes.
Definition 8. Consider an-IF submoduleof module; then,is called an-IF cyclic code having lengthoverif for eachwe have
Proposition 5. The-IF submoduleis an-IF cyclic code overif and only ifare cyclic codes over.
Proof. Let be an -IF submodule and be a cyclic code if is non-empty; then, for any , where we have As is a cyclic code, we havewhich imply that . Thus, is cyclic code.
Conversely, suppose that and is a cyclic code. If is not a cyclic code, then there is an element such that . Suppose . Similarly, imply that ; thus, is a cyclic code but , which is a contradiction, so is a cyclic code.
Proposition 6. Letbe a module andbe an-IF submodule.is an-IF cyclic code on moduleif and only if the characteristic function of a level setis an-IF cyclic code on.
Proposition 7. Consider a module; then,is an-IF cyclic code on moduleif and only if for eachwe have
Proof. Let be a module and be an -IF cyclic code on module ; then, we haveThus,Conversely, suppose that the above equality holds for and ; then, by Definition 8, is an -IF cyclic code.
Theorem 4. Consider two-IF cyclic codesand; then, we have the following:(1)is an-IF cyclic code.(2)is an-IF cyclic code.(3)is an-IF cyclic code.
Proof. (1)Let and be two -IF modules of module such that and are -IF cyclic codes corresponding to -IF modules. Then, for , In a similar manner, we have By taking intersection of two -IF modules, we get again an -IF module; thus, is an -IF cyclic code.(2)Now, for , Similarly, we can show . Thus, is an -IF cyclic code.(3)Proof follows from (2).
Proposition 8. Ifis a module, thenis said to be an-IF cyclic code iff the non-empty level setsare-IF ideals of the factor ring.
Proof. Consider a module and a factor ring . Define a mapping as which is also an isomorphism. Let such that .
Suppose that be an -IF cyclic code; this implies that is an -IF cyclic code over . Cyclic codes are ideals in factor ring, which implies that is ideal of factor ring.
Conversely, suppose that for , the set is non-empty; being an ideal of a factor ring implies that is a submodule. Thus, the level set is a linear code implying the linearity of . If we define mapping , then the level set is an -IF cyclic code; thus, is an -IF cyclic code over ring .
As is a finite ring and if is a submodule, then and are also finite; then, we have . Suppose is the generator polynomial for ; then, , .
Theorem 5. Letbe a set of polynomials such that the polynomial for each . Ifforand, then the set of polynomials determines an-IF cyclic code, whereis the collection of level cut cyclic subcodes of.
Proof. Proof follows from Proposition 8.
3.3. IF Gray Map
The gray code, which is also called reflected binary code, is an ordering of the binary numeral system such that two successive values vary in a single bit. We will define -IF gray code by using the compositions and . Consider a map