# An Analysis of Algebraic Codes over Lattice Valued Intuitionistic Fuzzy Type-3R-Submodules

Authors: Asra Riaz; Sajida Kousar; Nasreen Kausar; Dragan Pamucar; Gezahagne Mulat Addis
Journal: Computational Intelligence and Neuroscience
 (2022)

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## 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 α-intuitionistic fuzzy (α-IF) submodule. The proof that the level set of LIF-3 is contained in the level set of α-IF is given, and it is exclusively employed to define linear codes over α-IF submodule. Further, α-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 α-IF submodule is entirely applicable in this scenario. More specifically, a mapping from the ℤ64 module to a lattice L (comprising 64 codons) is considered, and α-IF codes are defined along with the respective degrees.


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## Body

## 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 fieldF2, among which the binary codes are the simplest. Say, in a binary field F2, 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 μAx=0.7 shows that x belongs to set A 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 X, as a function from subset X to a lattice L. Atanassov and Stoeva [8] proposed the definition of lattice valued intuitionistic fuzzy set (LIFS-1) where a complete lattice with a unary operation N:L⟶L is considered. Gerstenkorn and Tepa v⌣ cev i˙ [9] extended the concept formulated by Atanassov and defined LIFS-2 by replacing unary operation with unary operator N by a linearization function ℓ:L⟶0,1. The choice of linearization creates a problem; to overcome this drawback, it is replaced by Lattice homomorphism α:L⟶0,1, 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.Sˇesˇ elja and Tepav cˇ evi c´ [12] introduced another method of involving fuzzy theory in coding based on defining a map A¯ from a non-empty set S=1,2,…,n to partially ordered set P. S⌣es⌣ elja et al. [13] used the concept and defined binary block codes over lattice valued fuzzy sets (L-fuzzy sets). Z⌣iz⌣ ovi c˙ and Lazarevi c˙ [14] discussed the length and cardinality of block codes over L-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 F, 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 G-invariant codes from primitive permutation representations of Mathieu groups M24 and M23.In this article, LIFS-3 codes overR-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 L-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 L-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-3R-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

LetF be a finite field; then, Fn is an n-dimensional vector space over F. A code C over F is simply a subset of Fn. The elements of code C are called the codewords, and these codewords are represented as a1,a2,…,an. A code C is said to be linear code over field F if u,v∈C; then, αu+βv∈C for all α,β∈C. Thus, C is a subspace of Fn. If F=ℤ2=0,1 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 F be a finite field. A code C is called cyclic if the cyclic shift of each codeword in C is also a member in C. Let C be a code over F. Then, the corresponding dual code is defined to be(1)C⊥=u∈F:u.v=0,∀v∈C.Thus, the dual code consists of all the codewords that are orthogonal to every codeword inC.


### 2.2. Lattice Order Set

A relationϱ on a non-empty set is S is a subset of S×S. If x,y belongs to a relation ϱ, this implies x is related to y, or we can write it as xϱy. A relation on a set S is said to be a partial order relation if it satisfies three properties, that is, reflexive property x≤x,for allx∈X, antisymmetric property x≤y∧y≤ximplyx=y, and transitive property x≤y∧y≤zimplyx≤z.Consider two non-empty setsA and B, where B is a subset of A. An element a∈A is said to be the upper bound of B provided that(2)b≤afor allb∈B.In addition, the elementa is said to be lower bound of set B if(3)b≥afor allb∈B.IfU is the set of all upper bounds, then the least element of this set is called a supremum or join ∨ and if L is the set of all lower bounds, then the greatest element of set L 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 S, a fuzzy set A is an ordered set (element of universe, degree of that element). Mathematically,(4)A=x,μx:x∈S.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 universeS, 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(5)A=x,μx,νx:x∈S,with0≤μx+νx≤1.LetS be a non-empty set and L be a lattice. Consider f:S⟶L, g:S⟶L, where f and g are membership and nonmembership functions; then, a lattice valued intuitionistic fuzzy set of type-1 (LIFS-1) is the set S,L,f,g,N, where N is an involutive order reversing unary operator on L such that fx≤Ngx.LetS be a non-empty set and L be a lattice. Consider f:S⟶L, g:S⟶L; then, a lattice valued intuitionistic fuzzy set of type-2 (LIFS-2) is the set S,L,f,g,ℓ, where ℓ:L⟶0,1 is a linearization function satisfying ℓfx+ℓgx≤1.LetS be a non-empty set and L be a complete lattice with top element T and bottom element B. Consider f:S⟶L, g:S⟶L which are membership and nonmembership functions; then, a lattice valued intuitionistic fuzzy set of type-3 (LIFS-3) is the set S,L,f,g,α, where α:L⟶0,1 is a lattice homomorphism with αT=1, αB=0 satisfying(6)αℓ1∧ℓ2=αℓ1∧αℓ2,αℓ1∨ℓ2=αℓ1∨αℓ2∀ℓ1,ℓ2∈L,andαfs+αgs≤1 for all s∈S.


### 2.4.R-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 ringR is a set equipped with two binary operations, namely, addition + and multiplication ⋅ and satisfying the following axioms:(1)
R,+ is an Abelian group.(2)
R,⋅⟶ is a semigroup.(3)
Multiplication is distributive over addition; that is,a⋅b+c=a⋅b+a⋅c and b+c⋅a=b⋅a+c⋅a for all a,b,c∈R.The ringR is called commutative if a⋅b=b⋅a for all a,b∈R. Consider a commutative ring R. An R-module is a set M together with a binary operation addition ‘⊕’ and scalar multiplication ‘∗’, where ∗:R×M⟶M; then, for all r,s∈R, a,b∈M, we have the following:(1)
ra⊕b=ra⊕rb.(2)
r⊕sa=ra⊕sa.(3)
rs∗a=rs∗a.(4)
1Ra=a where 1R is the multiplication identity of the ring R.LetR=ℤpk be a ring; then, ℤpkn is an R-module.


## 2.1. Linear Codes

LetF be a finite field; then, Fn is an n-dimensional vector space over F. A code C over F is simply a subset of Fn. The elements of code C are called the codewords, and these codewords are represented as a1,a2,…,an. A code C is said to be linear code over field F if u,v∈C; then, αu+βv∈C for all α,β∈C. Thus, C is a subspace of Fn. If F=ℤ2=0,1 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 F be a finite field. A code C is called cyclic if the cyclic shift of each codeword in C is also a member in C. Let C be a code over F. Then, the corresponding dual code is defined to be(1)C⊥=u∈F:u.v=0,∀v∈C.Thus, the dual code consists of all the codewords that are orthogonal to every codeword inC.


## 2.2. Lattice Order Set

A relationϱ on a non-empty set is S is a subset of S×S. If x,y belongs to a relation ϱ, this implies x is related to y, or we can write it as xϱy. A relation on a set S is said to be a partial order relation if it satisfies three properties, that is, reflexive property x≤x,for allx∈X, antisymmetric property x≤y∧y≤ximplyx=y, and transitive property x≤y∧y≤zimplyx≤z.Consider two non-empty setsA and B, where B is a subset of A. An element a∈A is said to be the upper bound of B provided that(2)b≤afor allb∈B.In addition, the elementa is said to be lower bound of set B if(3)b≥afor allb∈B.IfU is the set of all upper bounds, then the least element of this set is called a supremum or join ∨ and if L is the set of all lower bounds, then the greatest element of set L 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 S, a fuzzy set A is an ordered set (element of universe, degree of that element). Mathematically,(4)A=x,μx:x∈S.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 universeS, 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(5)A=x,μx,νx:x∈S,with0≤μx+νx≤1.LetS be a non-empty set and L be a lattice. Consider f:S⟶L, g:S⟶L, where f and g are membership and nonmembership functions; then, a lattice valued intuitionistic fuzzy set of type-1 (LIFS-1) is the set S,L,f,g,N, where N is an involutive order reversing unary operator on L such that fx≤Ngx.LetS be a non-empty set and L be a lattice. Consider f:S⟶L, g:S⟶L; then, a lattice valued intuitionistic fuzzy set of type-2 (LIFS-2) is the set S,L,f,g,ℓ, where ℓ:L⟶0,1 is a linearization function satisfying ℓfx+ℓgx≤1.LetS be a non-empty set and L be a complete lattice with top element T and bottom element B. Consider f:S⟶L, g:S⟶L which are membership and nonmembership functions; then, a lattice valued intuitionistic fuzzy set of type-3 (LIFS-3) is the set S,L,f,g,α, where α:L⟶0,1 is a lattice homomorphism with αT=1, αB=0 satisfying(6)αℓ1∧ℓ2=αℓ1∧αℓ2,αℓ1∨ℓ2=αℓ1∨αℓ2∀ℓ1,ℓ2∈L,andαfs+αgs≤1 for all s∈S.


## 2.4.R-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 ringR is a set equipped with two binary operations, namely, addition + and multiplication ⋅ and satisfying the following axioms:(1)
R,+ is an Abelian group.(2)
R,⋅⟶ is a semigroup.(3)
Multiplication is distributive over addition; that is,a⋅b+c=a⋅b+a⋅c and b+c⋅a=b⋅a+c⋅a for all a,b,c∈R.The ringR is called commutative if a⋅b=b⋅a for all a,b∈R. Consider a commutative ring R. An R-module is a set M together with a binary operation addition ‘⊕’ and scalar multiplication ‘∗’, where ∗:R×M⟶M; then, for all r,s∈R, a,b∈M, we have the following:(1)
ra⊕b=ra⊕rb.(2)
r⊕sa=ra⊕sa.(3)
rs∗a=rs∗a.(4)
1Ra=a where 1R is the multiplication identity of the ring R.LetR=ℤpk be a ring; then, ℤpkn is an R-module.


## 3. Lattice Valued Intuitionistic Fuzzy Type-3R-Submodule

IfA=M,f,g,L,α is a lattice valued intuitionistic fuzzy subset of type-3, where f:M⟶L and g:M⟶L, then A is called an LIF-3 submodule of M if for r∈R and a,b∈M we have the following:(1)
fa+b≥fa∧fb.(2)
fr.a≥fa.(3)
ga+b≤ga∨gb.(4)
gr.a≤ga.Letfα,gα:M⟶0,1 be two composition functions, where fα=αof and gα=αog. Then, A¯=M,fα,gα is called an α-intuitionistic fuzzy (α-IF) submodule if for r∈R and x,y∈M we have the following:(1)
fαx+y≥fαx∧fαy.(2)
fαr.y≥fαx.(3)
gαx+y≤gαx∨gαy.(4)
gαr.y≤gαx.Remark 1.
IfA=M,f,g,L,αis an LIF-3 submodule, then(7)f0≥fx,g0≤gx.
Moreover, ifA¯=M,fα,gα is an α-IF submodule, then(8)fα0≥fαx,gα0≤gαx.Proposition 1.
LetA=M,f,g,L,αbe an LIF-3 submodule. Then, the necessary and sufficient conditions forAto be anR-module are as follows:(1)
fk1x+k2y≥fx∧fy,(2)
gk1x+k2y≤gx∨gy,
∀x,y∈M and k1,k2∈R.Definition 1.
An LIF-3 submoduleA=M,f,g,L,αof a ringRis called an LIF-3 ideal if for eachx,y∈R,(1)
fx−y≥fx∧fy.(2)
fx.y≥fx∨fy.(3)
gx−y≤gx∨gy.(4)
gx.y≥gx∧gy.Definition 2.
LetA=M,f,g,L,αbe an LIF-3 submodule; then, the level sets are defined as(9)Al1l2=m∈M:fm≥l1,gm≤l2.
In a similar fashion, we can define level sets for anα -IF submodule A¯=M,fα,gα as follows:(10)A¯t1t2=m∈M:fαm≥t1,gαm≤t2.
We can also write that(11)fαt1=m∈M:fαm≥t1,gαt2=m∈M:gαm≥t2.


### 3.1. LIF-3 Codes over Modules

If we consider a moduleM=ℤpkn which is a ℤpk-module, then an LIF-3 submodule A of M is termed as an LIF-3 linear code having length n over ℤpk.Proposition 2.
LetA=M,f,g,L,αbe an LIF-3 submodule andA¯=M,fα,gαbe anα-IF submodule. LetAl1l2andA¯t1t2be the level sets forAandA¯, whereαl1=t1andαl2=t2. Then,Al1l2⊆A¯t1t2.Proof.
LetAl1l2 and A¯t1t2 be two level sets for A and A¯. Let m∈Al1l2; then, fm≥l1 and gm≤l2 imply that fαm≥αl1 and gαm≤αl2, where αl1,αl2∈0,1. From this, we get that m∈Aαl1αl2. Thus, Al1l2⊆Aαl1αl2. Let αl1=t1 and αl2=t2; then, Al1l2⊆At1t2.
From the above proposition, we have concluded that the level set of LIF-3 submodule is contained inα-IF submodule, so we will use α-IF submodule for further discussion of codes. Let us consider a module M=ℤpkn which is a ℤpk module, and let A¯=M,fα,gα be an α-IF submodule; then, A¯ is said to be an α-IF linear code of length n over the module ℤpk.Example 1.
Considerℤ4-module and latticeL=0,1,a,b,c,d,having 0, 1 as bottom and top elements witha≤c,a≤d, andb≤c. Letf,g:ℤ4⟶Landα:L⟶0,1be defined as(12)f=01231abc,g=01230cda.Letα0=αb=0, αa=0.2=αc, and αd=α1=1. Then, A¯=M,fα,gα is an α -IF ℤ4 module; therefore, A¯ is an α -IF linear code.Definition 3.
Consider anα-IF submodule A¯=M,fα,gα; then, the number of the elements mapped in to the same element, say, t∈0,1, is said to be the degree of that element t and it is denoted by pt. In example 1, p0=2, p1=2, p0.2=2.Proposition 3.
Consider anα-IF submoduleA¯=M,fα,gα; A¯is anα-IF linear code overℤpkiffA¯t1t2≠ϕare linear codes.Definition 4.
LetA¯ be an α-IF submodule and k be non-empty subset of A¯. The α-IF characteristic function of k is denoted by χk=fαχk,gαχk, where fαχk,gαχk:M⟶0,1; then,(13)fαχkx=1,ifx∈k;0,ifx∉k,gαχkx=0,ifx∈k;1,ifx∉k.Theorem 1.
LetCbe a subset of anα-IF submoduleℤpkn; then,Cis anα-IF linear code overℤpkiffχCforCis anα-IF linear code over the same ringℤpk.Proof.
LetC be a linear code implying that C is an α-IF submodule. Now, if x,y∈C, then by Definition 4 of χC, we have(14)fαχCx=1=fαχCy,gαχCx=0=gαχCy.
AsC is an α-IF submodule, then x+y,r.x∈C and for all r∈R imply that fαχCx+y=1=1∧1=fαχCx∧fαχCy and fαχCr.x=1=fαχCx. Similarly, gαχCx+y≤gαχCx∨gαχCy and gαχCr.x≤gαχCx. Thus, the χC of C is an α-IF submodule; hence, it is an α-IF linear code.
Conversely, suppose that theχC be a linear code; hence, it is an α-IF submodule. Let x,y∈C; then, by Definition 4,(15)fαχCx=1=fαχCy,gαχCx=0=gαχCx,
imply thatfαχCx+y≥fαχCx∧fαχCy=1∧1=1 and fαχCr.x≥fαχCx=1. Similarly, gαχCx+y≤gαχCx∨gαχCy=0∨0=0 and gαχCr.x≥gαχCx=0; hence, x+y,r.x∈C; thus, C is an α-IF submodule, so C is an α-IF linear code.Proposition 4.
LetA¯be anα-IFS ofℤpkn. A¯is anα-IF linear code overℤpkiffχA¯t1t2is anα-IF linear code.Definition 5.
LetA¯=M,fα,gαbe anα-IF fuzzy submodule; then,A¯is also a linear code such that fort1,t2∈0,1, fαx=t1andgαx=t2; then,Ais said to be a trivialα-IF linear code.Definition 6.
LetM=ℤpknbe a module over the ringℤpk. Then, twoα-IF submodulesA¯=M,fα,gαandB¯=M,fα′,gα′are said to be orthogonal if(16)Imfα′=1−m:m∈Imfα,Imgα′=1−m′:m′∈Imgα.
Furthermore, for allt1,t2∈0,1, we have(17)fα1−t1=fαt1T=b∈M|a,b=0,for all a∈fαt1,gα1−t2′=gαt2T=b∈M:a,b=0for alla∈gαt2,where ⋅,⋅ is the inner product on module M.Example 2.
LetA¯,B¯:ℤ4⟶0,1be twoα-IF submodules which are defined as(18)fα=01230.50.250.330.25,gα=01230.1250.330.20.33,
and(19)fα′=01230.750.20.670.5,gα′=01230.670.8750.80.875.
Asfαt1=a∈M:fαa≥t1, we can compute the valuesfαt1fort1∈0,1.(20)fα0.5=a∈M:fαa≥0.5=0,fα0.25=a∈M:fαa≥0.25=0,1,2,3=ℤ4,fα0.33=a∈M:fαa≥0.33=0,2.
Similarly, the remaining values given in Table1 and2 can be obtained from the definition of level set.
Hence,A¯ is orthogonal to B¯. 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.Table 1
Values of orthogonal set for membership.
t1fαt1fα1−t1fαt1T0.50ℤ4ℤ40.25ℤ4000.330,20,20,20.25ℤ400Table 2
Values of orthogonal set for nonmembership.
t2gαt2gα1−t2gαt2T0.1250ℤ4ℤ40.33ℤ4000.20,20,20,20.33ℤ400Remark 2.
LetMbeℤpkmodule andA¯, B¯be twoα-IF submodules. If for allw∈M,fαw=rorgαw=r, wherer∈0,1, then setA¯is not orthogonal to setB¯.Remark 3.
LetA¯=M,fα,gαbe anα-IF submodule, soA¯is a linear code overℤpkhaving lengthn. Consider sets(21)imfα=fαx|x∈ℤpkn,imgα=fαx|x∈ℤpkn.
As the setℤpkn is finite, imfα and imgα are also finite. Consider an order t1≥t2≥⋯≥tr ; then, the set imfα satisfies this order. Similarly, suppose t1′≤t2′≤…≤tr′ and this order is satisfied by imgα. A¯t1t2 is a level set which is a linear code, where the generator matrix for this linear code is given by Gtjtj′ . Hence, A can be obtained from the matrices Gt1t1′,Gt2t2′,…,Gtrtr′.Theorem 2.
LetM=ℤpknbe a finite module andA¯=M,fα,gαbe anα-IF submodule ofM. Then, there is anα-IF submoduleB¯=M,fα′,gα′such thatA¯is orthogonal toB¯if and only ifImfα,Imgα>1, and for anyr∈Imfα, there exists an elementη∈Imfαwithfαr=fαηT. Similarly, for any elementp∈Imgα, there existð∈Imgαsuch thatgαp=gαðT.Proof.
LetM be a module and A¯=M,fα,gα be an α-IF submodule of M. Suppose that Imfα=s>1 and Imgα=k>1 and for any element r∈Imfα there is an element η∈Imfα such that fαr=fαηT. Moreover, for any element p∈Imgα, there exists an element ð∈Imgα such that gαp=gαðT.Now, consider an order for Imfα and Imgα as(22)Imfα=t1≥t2≥⋯≥tm,Imgα=t1′≤t2′≤⋯≤tm′.
By using these compositions, we can also define sets which form partition of moduleM as(23)Mj=a∈M:fαa=tj,Mj=a∈M:gαa=tj′,where j=1,…,m. Now, if we define an α-IF set B¯=M,fα′,gα′, where fα′,gα′:M⟶0,1, then for a∈M, we have fα′a=1−tm−j+1 and gα′a=1−tm−j+1′.
As we know,Imfα=t1≥t2≥⋯≥tm imply that fαt1⊆fαt2⊆⋯⊆fαtm. Similarly, for Imgα=t1′≤t2′≤⋯∈≤tm′, we have gαtm⊆⋯⊆gαt2⊆gαtm. As for any element r∈Imfα, there is an element η∈Imα°f such that fαr=α°fηT. Similarly, for any p∈Imgα, there is an element ð∈Imgα such that gαp=gαðT, as in finite module there is a property related to orthogonality by which fαtj=fαtm−j+1T. Accordingly, we have fα1−tm−j+1′=a∈M:fα′a≥1−tm−j+1=MjUMj−1U⋯UM1=fαtj=fαtm−j+1T.
Furthermore,gαtj′=gαtm−j+1′T and gα1−tm−j+1′′=a∈M:gα′a≥1−tm−j+1′=M1U…UMj−1UMj=gαtj=gαtm−j+1′T. Thus, B¯=M,fα′,gα′ is an α-IF submodule.
Conversely, suppose thatA¯=M,fα,gα and B¯=M,fα′,gα′ be two α-IF submodules such that these two sets are orthogonal to each other; then, we have Imfα>1 and Imgα>1. For all t,t′∈0,1, we also have fα1−t1′=fαtT and gα1−t1′′=gαt′T; then, for any element r∈Imfα, there exist η∈Imfα such that fαr=fαηT, and for any element p∈Imgα, there exist ð∈Imgα such that gαp=gαðT; this is due to the reason that imfα′=1−t:t∈fα and imgα′=1−t′:t′∈gα.Theorem 3.
Suppose thatA¯=M,fα,gα, B¯=M,fα′,gα′, andC¯=M,fα″,gα″be threeα-IF fuzzy submodules of a moduleMsuch thatA¯is orthogonal toB¯andB¯is orthogonal toC¯; then,B¯=C¯.Proof.
LetA¯=M,fα,gα, B¯=M,fα′,gα′, and C¯=M,fα″,gα″ be three α-IF fuzzy submodules of a module M such that A¯ is orthogonal to B¯ and B¯ is orthogonal to C¯. Let b∈fα1−t′, for t∈0,1. Then, for a∈fαt, a,b=0. B¯ is orthogonal to C¯, which implies that b∈fα1−t″. Thus, fα1−t′⊆fα1−t″. Therefore, fαt″⊆fαt′. By following the same strategy, we get fαt′⊆fαt″. Thus, fα′=fα″. In a similar manner, we can show that gα′ =  gα″. Hence, B¯=C¯.Corollary 1.
LetMbe a finiteℤpkmodule andA¯=M,fα,gαbe anα-IF submodule ofM; if there existsα-IFSB¯=M,fα′,gα′such thatB¯is orthogonal to the submoduleA¯, thenB¯is anα-IF submodule of moduleM.Definition 7.
Consider twoα-IF linear codesA¯=M,fα,gαandB¯=M,fα′,gα′; then, these twoα-IF codes are said to be equivalent if the level setsA¯t1t2andB¯t1t2(which are also linear codes) forA¯andB¯are equivalent.


### 3.2.α-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 discussα-IF cyclic codes and some important results related to these codes.Definition 8.
Consider anα-IF submoduleA¯=M,fα,gαof moduleℤpkn; then,A¯is called anα-IF cyclic code having lengthnoverℤpkif for eachw0,w1,…,wn−1∈ℤpknwe have(24)fαwn−1,w0,…,wn−2≥fαw0,w1,…,wn−1,gαwn−1,w0,…,wn−2≤gαw0,w1,…,wn−1.Proposition 5.
Theα-IF submoduleA¯=M,fα,gαis anα-IF cyclic code overℤpkif and only ifA¯t1t2≠∅are cyclic codes overℤpk.Proof.
LetA¯=M,fα,gα be an α-IF submodule and A¯ be a cyclic code if A¯t1t2 is non-empty; then, for any wi∈A¯t1t2, where i=0,1,…,n−1 we have (25)fαw0,w1,…,wn−1≥t1,gαw0,w1,…,wn−1≤t2.
AsA¯ is a cyclic code, we have(26)fαwn−1,w0,…,wn−2≥fαw0,w1,…,wn−1≥t1,gαwn−1,w0,…,wn−2≤gαw0,w1,…,wn−1≥t2,
which imply thatwn−1,w0,…,wn−2∈A¯t1t2. Thus, A¯t1t2 is cyclic code.
Conversely, suppose thatA¯t1t2≠ϕ and A¯t1t2 is a cyclic code. If A¯ is not a cyclic code, then there is an element w0,w1,…,wn−1∈ℤpkn such that fαwn−1,w0,…,wn−2<fαw0,w1,…,wn−1. Suppose t1′=fαw0,w1,…,wn−1. Similarly, t2′=gαw0,w1,…,wn−1 imply that w0,w1,…,wn−1∈A¯t1′t2′; thus, A¯t1′t2′≠ϕ is a cyclic code but wn−1,w0,…,wn−2∉A¯t1′t2′, which is a contradiction, so A¯ is a cyclic code.Proposition 6.
Letℤpknbe a module andA¯=M,fα,gαbe anα-IF submodule.A¯is anα-IF cyclic code on moduleℤpknif and only if the characteristic function of a level setA¯t1t2is anα-IF cyclic code onℤpkn.Proposition 7.
Consider a moduleℤpkn; then,A¯=M,fα,gαis anα-IF cyclic code on moduleℤpknif and only if for eachw0,w1,…,wn−1∈ℤpknwe have(27)fαw0,w1,…,wn−1=fαwn−1,w0,…,wn−2=⋯=fαw1,w2,…,wn−1,w0,gαw0,w1,…,wn−1=gαwn−1,w0,…,wn−2=⋯=gαw1,w2,…,wn−1,w0.Proof.
Letℤpkn be a module and A¯=M,fα,gα be an α-IF cyclic code on module ℤpkn; then, we have(28)fαw0,w1,…,wn−1≥fαwn−1,w0,…,wn−2≥⋯≥fαw1,w2,…,wn−1,w0≥fαw0,w1,…,wn−1,gαw0,w1,…,wn−1≤gαwn−1,w0,…,wn−2≤⋯≤gαw1,w2,…,wn−1,w0≤gαw0,w1,…,wn−1.
Thus,(29)fαw0,w1,…,wn−1=fαwn−1,w0,…,wn−2=⋯=fαw1,w2,…,wn−1,w0=fαw0,w1,…,wn−1,gαw0,w1,…,wn−1=gαwn−1,w0,…,wn−2=⋯=gαx1,w2,…,wn−1,w0=gαw0,w1,…,wn−1.
Conversely, suppose that the above equality holds forfα and gα; then, by Definition 8, A is an α-IF cyclic code.Theorem 4.
Consider twoα-IF cyclic codesA¯andB¯; then, we have the following:(1)
A¯∩B¯is anα-IF cyclic code.(2)
A¯+B¯is anα-IF cyclic code.(3)
A¯B¯is anα-IF cyclic code.Proof.
(1)
LetA¯=M,fα,gα and B¯=M,fα′,gα′ be two α-IF modules of module ℤpkn such that A¯ and B¯ are α-IF cyclic codes corresponding to α-IF modules. Then, for w0,w1,…,wn−1∈ℤpkn,(30)fα∩fα′wn−1,w0,…,wn−2=minfαwn−1,w0,…,wn−2,fα′wn−1,w0,…,wn−2≥minfαw0,w1,…,wn−1,fα′w0,w1,…,wn−1=fαw0,w1,…,wn−1∩fα′w0,w1,…,wn−1.In a similar manner, we have(31)gα∩gα′w0,w1,…,wn−1≤gαw0,w1,…,wn−1∩gα′w0,w1,…,wn−1.
By taking intersection of twoα-IF modules, we get again an α-IF module; thus, A¯∩B¯ is an α-IF cyclic code.(2)
Now, forw0,w1,…,wn−1∈ℤpkn,(32)fα+fα′wn−1,…,wn−2=maxjminyj+zj=wjfαyn−1,…,yn−2,fα′zn−1,…,zn−2≥maxminfαy0,y1,…,yn−1,fα′zn−1,z0,…,zn−1=fα+fα′w0,w1,…,wn−1.Similarly, we can showgα+gα′wn−1,w0,…,wn−2≤gα+gα′w0,w1,…,wn−1. Thus, A¯+B¯ is an α-IF cyclic code.(3)
Proof follows from (2).Proposition 8.
Ifℤpknis a module, thenA¯is said to be anα-IF cyclic code iff the non-empty level setsA¯t1t2areα-IF ideals of the factor ringℤpkX/Xn−1.Proof.
Consider a moduleℤpkn and a factor ring ℤpkX/Xn−1. Define a mapping as ϕ:ℤpkn⟶ZpkX/Xn−1 which is also an isomorphism. Let b=b0,b1,…,bn−1∈ℤpkn such that ϕb=∑j=0m−1bjXj.
Suppose thatA¯ be an α-IF cyclic code; this implies that A¯t1t2≠∅ is an α-IF cyclic code over ℤpk. Cyclic codes are ideals in factor ring, which implies that A¯t1t2 is ideal of factor ring.
Conversely, suppose that fort1,t2∈0,1, the set A¯t1t2 is non-empty; A¯t1t2 being an ideal of a factor ring implies that A¯t1t2 is a submodule. Thus, the level set A¯t1t2 is a linear code implying the linearity of A¯. If we define mapping ϕ, then the level set A¯t1t2 is an α-IF cyclic code; thus, A¯ is an α-IF cyclic code over ring ℤpk.
Asℤpk is a finite ring and if A¯=M,fα,gα is a submodule, then imfα and imgα are also finite; then, we have A¯t1t1′⊆A¯t2t2′⊆⋯⊆A¯tr−1tr−1′⊆A¯t2t2′=ℤpkn. Suppose gjkX∈ℤpkX is the generator polynomial for A¯titi′; then, gj+1kX/gjkX, j=1,…,r−1.Theorem 5.
LetS=g1kX,g2kX,…,grkXbe a set of polynomials such that the polynomial gjXdividesXn−1 for each j=1,…,r. Ifgj+1kX|gjkXforj=1,2,…,r−1andℤpkn=gj+1kX, then the set of polynomials determines anα-IF cyclic codeA¯, wheregjkXis the collection of level cut cyclic subcodes ofA¯.Proof.
Proof follows from Proposition8.


### 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 defineα-IF gray code by using the compositions fα and gα. Consider a map η:ℤ22⟶ℤ22; then, η is called a gray map defined as η0=00, η1=01, η2=11, η3=10.Definition 9.
Consider a mappingη:ℤ22⟶ℤ22which is a gray map. Suppose thatSℤ22andSℤ22be twoα-IF fuzzy subsets ofℤ22andℤ22. Forfα,gα∈Sℤ22, anα-IF gray mapη∗:Sℤ22⟶Sℤ22is defined as(33)η∗fα∧gαb=supfα∧gαa|b=ηa.Example 3.
Consider two mappingsfα,gα:ℤ4⟶0,1defined as follows:(34)fα=012310.40.40.4,gα=012300.30.30.3.
By definition, we have(35)η∗fα∧gαb=supfα∧gαa|b=ηa.
Now, fora=0,(36)η∗fα∧gαb=supfα∧gα0|b=η0η∗fα∧gα00=supfα∧gα0|00=η0=supfα0∧gα0|00=η0=sup1∧0=0.
The values forα-IF gray map corresponding to these mappings are given in Table 3.
Now, we consider generalizing fuzzy gray map which is a mapping fromℤpk to ℤppk−1. Using this map, an α-IF generalized gray map can be defined.Table 3
Values of gray map andα-IF gray map.
ηa=bη∗fα∧gαb000010.3110.3100.3Definition 10.
A mappingη∗:Sℤpk⟶Sℤppk−1is said to be anα-IF generalized gray map if for anyfα,gα∈Sℤpkwe have(37)η∗fα∧gαb=fα∧gαa,ifb=ηa;0,otherwise.Definition 11.
IfA¯is anα-IF code overℤpk, then it is said to be anα-IFℤpk-linear code if it is an image under theα-IF generalized gray map of a linear code overℤpk.Definition 12.
Consider anα-IF codeC; then, it is anα-IFℤpkcyclic code ifCis anα-IFℤpk-linear code and also if it is the image under theα-IF generalized gray map of a cyclic code overℤpk.Example 4.
Consider mappingsfα,gα:ℤ26⟶0,1where(38)fαx0,x1,x2,x3,x4,x5=1,ifx4=x5=0;0,otherwise,
and(39)gαx0,x1,x2,x3,x4,x5=0,ifx4=x5=0;1,otherwise.
Then,C=M,fα,gα is an α -IF linear code having length six over ℤ2 . Now, if we consider another code where fα,gα:ℤ43⟶0,1 defined asfα′x0,x1,x2=1,ifx2=0;0,otherwise, and gα′x0,x1,x2=0,ifx2=0;1,otherwise; then, C=M,fα,gα is an α -IF linear code having length 3 over ℤ4.Theorem 6.
Anα-IF gray mapη∗is a bijective map.Proof.
This is due to the fact thatη∗ is a one-to-one function.


### 3.4. Error Correction

The coding theory deals with encoding and decoding a message. During transmission of data, errors may occur; they can be detected and corrected by using different procedures such as parity check, syndrome decoding, and redundancy check which are for ordinary codes. Similarly, a method can be adopted in fuzzy codes for the confirmation about the codeword, that is, whether it belongs to the transmitted code or not. This can be done by considering the level set from which one can determine the confirmation degree as to whether a received codeword belongs to the original code. This method can be explained through the following example.Example 5.
Consider a moduleℤ24, where(40)ℤ24=0000,1000,0100,0010,0001,1100,1010,1001,0101,0110,0011,1011,1101,1110,0111,111.
LetA¯=M,fα,gα be an α-IF submodule where fα,gα:ℤ24⟶0,1 and(41)C=0000,0001,0010,1100,0011,1101,1110,1111,
which is a subset ofℤ24and also a linear code. Now, by transmitting this code, the received codeword has errors; assume that the received codewords are000;.01,010;0.001,011;0.01,100;0.1,101;0.01,110;\\0.1,001;0.01,111;0.999.
Table4 shows the values of fα,gα corresponding to the codewords of code C.
The level set is defined asA¯t1t2=a∈M:αofa≥t1,αoga≤t2; from this, we get for t1>0.9 and t2≤0.01 that the received codeword is in C.Table 4
Recovering of codewords
afαgαafαgαafαgαafαgα00001010000.90.101000.90.100100.990.0100010.990.0111000.990.0110100.90.110010.90.101010.90.101100.90.100110.990.0110110.90.111010.990.0111100.990.0101110.90.111110.990.01


### 3.5. Application

#### 3.5.1. Sphere inα-IF Linear Code

Supposeℤpkm be a module and A¯=M,fα,gα be an α-IF submodule. If C is an α-IF linear code, then the level set Ct1t2 of C is also an α-IF linear code. As the elements in ℤpkm are the words of length m over the alphabet 0,1,2,…,pk−1 and the level set Ct1t2 also consists of the elements of ℤpkm, then for a member a∈Ct1t2 and for any integer r≥0, the sphere of radius r and center a is defined as follows.Theorem 7.
Consider a moduleM=ℤpkm; then, the sphere having radiusrwhere0≤r≤mcontains(42)m0+m1pk−1+m2pk−12+⋯+mrpk−1r,
words.Proof.
Consider a moduleℤpkm. Let x be a fixed word in ℤpkm. Then, the number of the words which differ from x in m position is mwpk−1w.Example 6.
LetM=ℤ22, whereℤ22=00,01,10,11. Here,k=1, p=2, m=2, and0≤r≤2. If we take a wordx=10, then its distances from the remaining codewords ared10,00=1, d10,01=2, d10,11=1, and the sum of these distances is equal to 4. Now,20+212−1+222−12=1+2+1=4, which is also equal to 4.


### 3.6. Genetic Code with Moduleℤ64

Amino acid sequence of a protein can be determined from the sequence or order of DNA and RNA molecules [26]. Although the information about protein sequence is found in the sequence of nucleotides in DNA, we cannot say that the proteins formed directly from the DNA. RNA molecules are also involved in the formation of proteins. RNA is composed of four bases which are adenine (A), guanine (G), cytosine (C), and uracil (U). One unit is formed from the three adjacent bases which are called a codon which codes for an amino acid. For instance, the amino acid methionine is encoded by the codon AUG. There are 64 codons; 20 amino acids are specified by the 61 codons that make up proteins. The remaining three codons do not code for any amino acid.There is another important aspect related to genetic code which is the algebraic structure of a genetic code from which some important information can be obtained [27]. Many scientists [28, 29] have studied the algebraic structure of codons. They explore various methods to model the genetic code mathematically and use binary representation for the four bases. Nemzer [30] established binary representation of genetic codes. Press et al. [31] described the HEDGES (Hash Encoded, Decoded by Greedy Exhaustive Search) error-correcting code to repair basic DNA errors. Rocha et al. [32] analyzed the DNA sequence generated by linear codes over the ring ℤ4. Bennenni et al. [33], Dinh et al. [34], and Gowthaman et al. [35] investigated DNA cyclic codes over rings.In the presented study, codon structure is investigated by involving the lattice structure of codons along with module over a ring. As there are 64 possible codons, we takeℤ64 module. From this, we conclude that α-IF submodule with module ℤ64 is a linear code. Consider a module M=ℤ64 and let fα,gα:ℤ64⟶0,1. Suppose f,g:ℤ64⟶L where L is a lattice consisting of 64 codons; the values for these functions and their composition are shown in Table 5. Here, A¯=M,fα,gα is an α-IF submodule so it is an α-IF linear code. We will use the following methodology.Table 5
Codons along with their respective degrees.
af(a)gaa(f(a))0UUUAAA11UCUAGA0.352UUGCAA0.43UCGAGC0.34GGUAAC0.45GCUCGA0.36UUCGAA0.357UCCAGG0.288GUGCAC0.359GCGCGC0.2810CUUAAG0.3511CCUGGA0.2812UUAUAA0.313UCAAGU0.2814GUCGAC0.315GCCCGG0.2516CUGCAG0.317CCGGGC0.2518AUUAAU0.319ACUUGA0.2520GUAUAC0.2821GCACGU0.222CUCGAG0.2823CCCGGG0.224AUGCAU0.2825ACGUGC0.226CUAUAG0.2527CCAGGU0.3528AUCGAU0.2529ACCUGG0.1530AUAUAU0.2531ACAUGU0.132UGUACA0.433UAUAUA0.334UGGACC0.3535UAGAUC0.2836GGUCCA0.3537GAUCUA0.2838UGCACG0.339UACAUG0.2540GGGCCC0.341GAGCUC0.2542CGUGCA0.343CAUGUA0.2544UGAACU0.2845UAAAUU0.2846GGCCCG0.2847GACCUG0.2848CGGGCC0.2849CAGGUC0.2850AGUUCA0.2851AAUUUA0.2852GGACCU0.2553GAACUU0.254CGCGCG0.2555CACGUG0.256AGGUCC0.2557AAGUUC0.258CGAGCU0.259CAAGUU0.1560AGCUCG0.261ACCUUG0.1562AGAUCU0.1563AAAUUU0Consider the latticeL comprises 64 codons and a module Z64=0,1,…,63 which also consists of 64 elements. Let a=0∈Z64; we know that f,g:Z64⟶L. Then, f0=UUU,g0=AAA, which are top and bottom elements of the lattice L as shown in [36]. As fα,gα:ℤ64⟶0,1, assign the values to these functions as fα0=1∈0,1, gα0=0∈0,1, where fα0+gα0≤1. Similarly, values can be assigned to the remaining codons, which is shown in Table 5. Then, by Definition 3, α-IF submodule with module Z64 is referred to as a linear code. By using Definition 4, we can compute the degrees of element t∈0,1; for example, for t=1, the degree is 1 and for t=0.3 the degree is 10. The remaining values can be computed in a similar way. Degrees of respective codons are also given in Table 5.The most important properties of amino acids are their hydrophobic and hydrophilic properties. As in lattice diagram, codons with second baseU show hydrophobic amino acids, and those with second base A show hydrophilic amino acids. There are total 16 codons that specify hydrophobic and hydrophilic amino acids, so we can define an α-IF linear code by considering the ℤ16 module. Let f,g:ℤ16⟶L and fα,gα:ℤ16⟶0,1, where lattice L consists of 16 elements. A comparison can be made between the two properties of amino acid by employing the concept of degree given in Definition 4. It can be seen in Table 6 that the average degree of hydrophobic amino acids is greater than the hydrophilic amino acids.Table 6
Codons withU or A as a second base which are coding for hydrophobic or hydrophilic amino acid.
afafαaDegreegagαaDegreeafafαaDegreegagαaDegree0UUU11AAA011AUA0.11AAC0.0122AUC0.122CAA0.0123CUA0.122AAG0.0534AUG0.23CAC0.0535CUC0.23GAA0.0536GUA0.23AAU0.147AUU0.254CAG0.148CUG0.254GAC0.149GUC0.254UAA0.1410UUA0.254CAU0.12211CUU0.33GAG0.12212GUG0.33UAC0.12213UUC0.33GAU0.2214GUU0.352UAG0.2215UUG0.352UAU0.251


## 3.1. LIF-3 Codes over Modules

If we consider a moduleM=ℤpkn which is a ℤpk-module, then an LIF-3 submodule A of M is termed as an LIF-3 linear code having length n over ℤpk.Proposition 2.
LetA=M,f,g,L,αbe an LIF-3 submodule andA¯=M,fα,gαbe anα-IF submodule. LetAl1l2andA¯t1t2be the level sets forAandA¯, whereαl1=t1andαl2=t2. Then,Al1l2⊆A¯t1t2.Proof.
LetAl1l2 and A¯t1t2 be two level sets for A and A¯. Let m∈Al1l2; then, fm≥l1 and gm≤l2 imply that fαm≥αl1 and gαm≤αl2, where αl1,αl2∈0,1. From this, we get that m∈Aαl1αl2. Thus, Al1l2⊆Aαl1αl2. Let αl1=t1 and αl2=t2; then, Al1l2⊆At1t2.
From the above proposition, we have concluded that the level set of LIF-3 submodule is contained inα-IF submodule, so we will use α-IF submodule for further discussion of codes. Let us consider a module M=ℤpkn which is a ℤpk module, and let A¯=M,fα,gα be an α-IF submodule; then, A¯ is said to be an α-IF linear code of length n over the module ℤpk.Example 1.
Considerℤ4-module and latticeL=0,1,a,b,c,d,having 0, 1 as bottom and top elements witha≤c,a≤d, andb≤c. Letf,g:ℤ4⟶Landα:L⟶0,1be defined as(12)f=01231abc,g=01230cda.Letα0=αb=0, αa=0.2=αc, and αd=α1=1. Then, A¯=M,fα,gα is an α -IF ℤ4 module; therefore, A¯ is an α -IF linear code.Definition 3.
Consider anα-IF submodule A¯=M,fα,gα; then, the number of the elements mapped in to the same element, say, t∈0,1, is said to be the degree of that element t and it is denoted by pt. In example 1, p0=2, p1=2, p0.2=2.Proposition 3.
Consider anα-IF submoduleA¯=M,fα,gα; A¯is anα-IF linear code overℤpkiffA¯t1t2≠ϕare linear codes.Definition 4.
LetA¯ be an α-IF submodule and k be non-empty subset of A¯. The α-IF characteristic function of k is denoted by χk=fαχk,gαχk, where fαχk,gαχk:M⟶0,1; then,(13)fαχkx=1,ifx∈k;0,ifx∉k,gαχkx=0,ifx∈k;1,ifx∉k.Theorem 1.
LetCbe a subset of anα-IF submoduleℤpkn; then,Cis anα-IF linear code overℤpkiffχCforCis anα-IF linear code over the same ringℤpk.Proof.
LetC be a linear code implying that C is an α-IF submodule. Now, if x,y∈C, then by Definition 4 of χC, we have(14)fαχCx=1=fαχCy,gαχCx=0=gαχCy.
AsC is an α-IF submodule, then x+y,r.x∈C and for all r∈R imply that fαχCx+y=1=1∧1=fαχCx∧fαχCy and fαχCr.x=1=fαχCx. Similarly, gαχCx+y≤gαχCx∨gαχCy and gαχCr.x≤gαχCx. Thus, the χC of C is an α-IF submodule; hence, it is an α-IF linear code.
Conversely, suppose that theχC be a linear code; hence, it is an α-IF submodule. Let x,y∈C; then, by Definition 4,(15)fαχCx=1=fαχCy,gαχCx=0=gαχCx,
imply thatfαχCx+y≥fαχCx∧fαχCy=1∧1=1 and fαχCr.x≥fαχCx=1. Similarly, gαχCx+y≤gαχCx∨gαχCy=0∨0=0 and gαχCr.x≥gαχCx=0; hence, x+y,r.x∈C; thus, C is an α-IF submodule, so C is an α-IF linear code.Proposition 4.
LetA¯be anα-IFS ofℤpkn. A¯is anα-IF linear code overℤpkiffχA¯t1t2is anα-IF linear code.Definition 5.
LetA¯=M,fα,gαbe anα-IF fuzzy submodule; then,A¯is also a linear code such that fort1,t2∈0,1, fαx=t1andgαx=t2; then,Ais said to be a trivialα-IF linear code.Definition 6.
LetM=ℤpknbe a module over the ringℤpk. Then, twoα-IF submodulesA¯=M,fα,gαandB¯=M,fα′,gα′are said to be orthogonal if(16)Imfα′=1−m:m∈Imfα,Imgα′=1−m′:m′∈Imgα.
Furthermore, for allt1,t2∈0,1, we have(17)fα1−t1=fαt1T=b∈M|a,b=0,for all a∈fαt1,gα1−t2′=gαt2T=b∈M:a,b=0for alla∈gαt2,where ⋅,⋅ is the inner product on module M.Example 2.
LetA¯,B¯:ℤ4⟶0,1be twoα-IF submodules which are defined as(18)fα=01230.50.250.330.25,gα=01230.1250.330.20.33,
and(19)fα′=01230.750.20.670.5,gα′=01230.670.8750.80.875.
Asfαt1=a∈M:fαa≥t1, we can compute the valuesfαt1fort1∈0,1.(20)fα0.5=a∈M:fαa≥0.5=0,fα0.25=a∈M:fαa≥0.25=0,1,2,3=ℤ4,fα0.33=a∈M:fαa≥0.33=0,2.
Similarly, the remaining values given in Table1 and2 can be obtained from the definition of level set.
Hence,A¯ is orthogonal to B¯. 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.Table 1
Values of orthogonal set for membership.
t1fαt1fα1−t1fαt1T0.50ℤ4ℤ40.25ℤ4000.330,20,20,20.25ℤ400Table 2
Values of orthogonal set for nonmembership.
t2gαt2gα1−t2gαt2T0.1250ℤ4ℤ40.33ℤ4000.20,20,20,20.33ℤ400Remark 2.
LetMbeℤpkmodule andA¯, B¯be twoα-IF submodules. If for allw∈M,fαw=rorgαw=r, wherer∈0,1, then setA¯is not orthogonal to setB¯.Remark 3.
LetA¯=M,fα,gαbe anα-IF submodule, soA¯is a linear code overℤpkhaving lengthn. Consider sets(21)imfα=fαx|x∈ℤpkn,imgα=fαx|x∈ℤpkn.
As the setℤpkn is finite, imfα and imgα are also finite. Consider an order t1≥t2≥⋯≥tr ; then, the set imfα satisfies this order. Similarly, suppose t1′≤t2′≤…≤tr′ and this order is satisfied by imgα. A¯t1t2 is a level set which is a linear code, where the generator matrix for this linear code is given by Gtjtj′ . Hence, A can be obtained from the matrices Gt1t1′,Gt2t2′,…,Gtrtr′.Theorem 2.
LetM=ℤpknbe a finite module andA¯=M,fα,gαbe anα-IF submodule ofM. Then, there is anα-IF submoduleB¯=M,fα′,gα′such thatA¯is orthogonal toB¯if and only ifImfα,Imgα>1, and for anyr∈Imfα, there exists an elementη∈Imfαwithfαr=fαηT. Similarly, for any elementp∈Imgα, there existð∈Imgαsuch thatgαp=gαðT.Proof.
LetM be a module and A¯=M,fα,gα be an α-IF submodule of M. Suppose that Imfα=s>1 and Imgα=k>1 and for any element r∈Imfα there is an element η∈Imfα such that fαr=fαηT. Moreover, for any element p∈Imgα, there exists an element ð∈Imgα such that gαp=gαðT.Now, consider an order for Imfα and Imgα as(22)Imfα=t1≥t2≥⋯≥tm,Imgα=t1′≤t2′≤⋯≤tm′.
By using these compositions, we can also define sets which form partition of moduleM as(23)Mj=a∈M:fαa=tj,Mj=a∈M:gαa=tj′,where j=1,…,m. Now, if we define an α-IF set B¯=M,fα′,gα′, where fα′,gα′:M⟶0,1, then for a∈M, we have fα′a=1−tm−j+1 and gα′a=1−tm−j+1′.
As we know,Imfα=t1≥t2≥⋯≥tm imply that fαt1⊆fαt2⊆⋯⊆fαtm. Similarly, for Imgα=t1′≤t2′≤⋯∈≤tm′, we have gαtm⊆⋯⊆gαt2⊆gαtm. As for any element r∈Imfα, there is an element η∈Imα°f such that fαr=α°fηT. Similarly, for any p∈Imgα, there is an element ð∈Imgα such that gαp=gαðT, as in finite module there is a property related to orthogonality by which fαtj=fαtm−j+1T. Accordingly, we have fα1−tm−j+1′=a∈M:fα′a≥1−tm−j+1=MjUMj−1U⋯UM1=fαtj=fαtm−j+1T.
Furthermore,gαtj′=gαtm−j+1′T and gα1−tm−j+1′′=a∈M:gα′a≥1−tm−j+1′=M1U…UMj−1UMj=gαtj=gαtm−j+1′T. Thus, B¯=M,fα′,gα′ is an α-IF submodule.
Conversely, suppose thatA¯=M,fα,gα and B¯=M,fα′,gα′ be two α-IF submodules such that these two sets are orthogonal to each other; then, we have Imfα>1 and Imgα>1. For all t,t′∈0,1, we also have fα1−t1′=fαtT and gα1−t1′′=gαt′T; then, for any element r∈Imfα, there exist η∈Imfα such that fαr=fαηT, and for any element p∈Imgα, there exist ð∈Imgα such that gαp=gαðT; this is due to the reason that imfα′=1−t:t∈fα and imgα′=1−t′:t′∈gα.Theorem 3.
Suppose thatA¯=M,fα,gα, B¯=M,fα′,gα′, andC¯=M,fα″,gα″be threeα-IF fuzzy submodules of a moduleMsuch thatA¯is orthogonal toB¯andB¯is orthogonal toC¯; then,B¯=C¯.Proof.
LetA¯=M,fα,gα, B¯=M,fα′,gα′, and C¯=M,fα″,gα″ be three α-IF fuzzy submodules of a module M such that A¯ is orthogonal to B¯ and B¯ is orthogonal to C¯. Let b∈fα1−t′, for t∈0,1. Then, for a∈fαt, a,b=0. B¯ is orthogonal to C¯, which implies that b∈fα1−t″. Thus, fα1−t′⊆fα1−t″. Therefore, fαt″⊆fαt′. By following the same strategy, we get fαt′⊆fαt″. Thus, fα′=fα″. In a similar manner, we can show that gα′ =  gα″. Hence, B¯=C¯.Corollary 1.
LetMbe a finiteℤpkmodule andA¯=M,fα,gαbe anα-IF submodule ofM; if there existsα-IFSB¯=M,fα′,gα′such thatB¯is orthogonal to the submoduleA¯, thenB¯is anα-IF submodule of moduleM.Definition 7.
Consider twoα-IF linear codesA¯=M,fα,gαandB¯=M,fα′,gα′; then, these twoα-IF codes are said to be equivalent if the level setsA¯t1t2andB¯t1t2(which are also linear codes) forA¯andB¯are equivalent.


## 3.2.α-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 discussα-IF cyclic codes and some important results related to these codes.Definition 8.
Consider anα-IF submoduleA¯=M,fα,gαof moduleℤpkn; then,A¯is called anα-IF cyclic code having lengthnoverℤpkif for eachw0,w1,…,wn−1∈ℤpknwe have(24)fαwn−1,w0,…,wn−2≥fαw0,w1,…,wn−1,gαwn−1,w0,…,wn−2≤gαw0,w1,…,wn−1.Proposition 5.
Theα-IF submoduleA¯=M,fα,gαis anα-IF cyclic code overℤpkif and only ifA¯t1t2≠∅are cyclic codes overℤpk.Proof.
LetA¯=M,fα,gα be an α-IF submodule and A¯ be a cyclic code if A¯t1t2 is non-empty; then, for any wi∈A¯t1t2, where i=0,1,…,n−1 we have (25)fαw0,w1,…,wn−1≥t1,gαw0,w1,…,wn−1≤t2.
AsA¯ is a cyclic code, we have(26)fαwn−1,w0,…,wn−2≥fαw0,w1,…,wn−1≥t1,gαwn−1,w0,…,wn−2≤gαw0,w1,…,wn−1≥t2,
which imply thatwn−1,w0,…,wn−2∈A¯t1t2. Thus, A¯t1t2 is cyclic code.
Conversely, suppose thatA¯t1t2≠ϕ and A¯t1t2 is a cyclic code. If A¯ is not a cyclic code, then there is an element w0,w1,…,wn−1∈ℤpkn such that fαwn−1,w0,…,wn−2<fαw0,w1,…,wn−1. Suppose t1′=fαw0,w1,…,wn−1. Similarly, t2′=gαw0,w1,…,wn−1 imply that w0,w1,…,wn−1∈A¯t1′t2′; thus, A¯t1′t2′≠ϕ is a cyclic code but wn−1,w0,…,wn−2∉A¯t1′t2′, which is a contradiction, so A¯ is a cyclic code.Proposition 6.
Letℤpknbe a module andA¯=M,fα,gαbe anα-IF submodule.A¯is anα-IF cyclic code on moduleℤpknif and only if the characteristic function of a level setA¯t1t2is anα-IF cyclic code onℤpkn.Proposition 7.
Consider a moduleℤpkn; then,A¯=M,fα,gαis anα-IF cyclic code on moduleℤpknif and only if for eachw0,w1,…,wn−1∈ℤpknwe have(27)fαw0,w1,…,wn−1=fαwn−1,w0,…,wn−2=⋯=fαw1,w2,…,wn−1,w0,gαw0,w1,…,wn−1=gαwn−1,w0,…,wn−2=⋯=gαw1,w2,…,wn−1,w0.Proof.
Letℤpkn be a module and A¯=M,fα,gα be an α-IF cyclic code on module ℤpkn; then, we have(28)fαw0,w1,…,wn−1≥fαwn−1,w0,…,wn−2≥⋯≥fαw1,w2,…,wn−1,w0≥fαw0,w1,…,wn−1,gαw0,w1,…,wn−1≤gαwn−1,w0,…,wn−2≤⋯≤gαw1,w2,…,wn−1,w0≤gαw0,w1,…,wn−1.
Thus,(29)fαw0,w1,…,wn−1=fαwn−1,w0,…,wn−2=⋯=fαw1,w2,…,wn−1,w0=fαw0,w1,…,wn−1,gαw0,w1,…,wn−1=gαwn−1,w0,…,wn−2=⋯=gαx1,w2,…,wn−1,w0=gαw0,w1,…,wn−1.
Conversely, suppose that the above equality holds forfα and gα; then, by Definition 8, A is an α-IF cyclic code.Theorem 4.
Consider twoα-IF cyclic codesA¯andB¯; then, we have the following:(1)
A¯∩B¯is anα-IF cyclic code.(2)
A¯+B¯is anα-IF cyclic code.(3)
A¯B¯is anα-IF cyclic code.Proof.
(1)
LetA¯=M,fα,gα and B¯=M,fα′,gα′ be two α-IF modules of module ℤpkn such that A¯ and B¯ are α-IF cyclic codes corresponding to α-IF modules. Then, for w0,w1,…,wn−1∈ℤpkn,(30)fα∩fα′wn−1,w0,…,wn−2=minfαwn−1,w0,…,wn−2,fα′wn−1,w0,…,wn−2≥minfαw0,w1,…,wn−1,fα′w0,w1,…,wn−1=fαw0,w1,…,wn−1∩fα′w0,w1,…,wn−1.In a similar manner, we have(31)gα∩gα′w0,w1,…,wn−1≤gαw0,w1,…,wn−1∩gα′w0,w1,…,wn−1.
By taking intersection of twoα-IF modules, we get again an α-IF module; thus, A¯∩B¯ is an α-IF cyclic code.(2)
Now, forw0,w1,…,wn−1∈ℤpkn,(32)fα+fα′wn−1,…,wn−2=maxjminyj+zj=wjfαyn−1,…,yn−2,fα′zn−1,…,zn−2≥maxminfαy0,y1,…,yn−1,fα′zn−1,z0,…,zn−1=fα+fα′w0,w1,…,wn−1.Similarly, we can showgα+gα′wn−1,w0,…,wn−2≤gα+gα′w0,w1,…,wn−1. Thus, A¯+B¯ is an α-IF cyclic code.(3)
Proof follows from (2).Proposition 8.
Ifℤpknis a module, thenA¯is said to be anα-IF cyclic code iff the non-empty level setsA¯t1t2areα-IF ideals of the factor ringℤpkX/Xn−1.Proof.
Consider a moduleℤpkn and a factor ring ℤpkX/Xn−1. Define a mapping as ϕ:ℤpkn⟶ZpkX/Xn−1 which is also an isomorphism. Let b=b0,b1,…,bn−1∈ℤpkn such that ϕb=∑j=0m−1bjXj.
Suppose thatA¯ be an α-IF cyclic code; this implies that A¯t1t2≠∅ is an α-IF cyclic code over ℤpk. Cyclic codes are ideals in factor ring, which implies that A¯t1t2 is ideal of factor ring.
Conversely, suppose that fort1,t2∈0,1, the set A¯t1t2 is non-empty; A¯t1t2 being an ideal of a factor ring implies that A¯t1t2 is a submodule. Thus, the level set A¯t1t2 is a linear code implying the linearity of A¯. If we define mapping ϕ, then the level set A¯t1t2 is an α-IF cyclic code; thus, A¯ is an α-IF cyclic code over ring ℤpk.
Asℤpk is a finite ring and if A¯=M,fα,gα is a submodule, then imfα and imgα are also finite; then, we have A¯t1t1′⊆A¯t2t2′⊆⋯⊆A¯tr−1tr−1′⊆A¯t2t2′=ℤpkn. Suppose gjkX∈ℤpkX is the generator polynomial for A¯titi′; then, gj+1kX/gjkX, j=1,…,r−1.Theorem 5.
LetS=g1kX,g2kX,…,grkXbe a set of polynomials such that the polynomial gjXdividesXn−1 for each j=1,…,r. Ifgj+1kX|gjkXforj=1,2,…,r−1andℤpkn=gj+1kX, then the set of polynomials determines anα-IF cyclic codeA¯, wheregjkXis the collection of level cut cyclic subcodes ofA¯.Proof.
Proof follows from Proposition8.


## 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 defineα-IF gray code by using the compositions fα and gα. Consider a map η:ℤ22⟶ℤ22; then, η is called a gray map defined as η0=00, η1=01, η2=11, η3=10.Definition 9.
Consider a mappingη:ℤ22⟶ℤ22which is a gray map. Suppose thatSℤ22andSℤ22be twoα-IF fuzzy subsets ofℤ22andℤ22. Forfα,gα∈Sℤ22, anα-IF gray mapη∗:Sℤ22⟶Sℤ22is defined as(33)η∗fα∧gαb=supfα∧gαa|b=ηa.Example 3.
Consider two mappingsfα,gα:ℤ4⟶0,1defined as follows:(34)fα=012310.40.40.4,gα=012300.30.30.3.
By definition, we have(35)η∗fα∧gαb=supfα∧gαa|b=ηa.
Now, fora=0,(36)η∗fα∧gαb=supfα∧gα0|b=η0η∗fα∧gα00=supfα∧gα0|00=η0=supfα0∧gα0|00=η0=sup1∧0=0.
The values forα-IF gray map corresponding to these mappings are given in Table 3.
Now, we consider generalizing fuzzy gray map which is a mapping fromℤpk to ℤppk−1. Using this map, an α-IF generalized gray map can be defined.Table 3
Values of gray map andα-IF gray map.
ηa=bη∗fα∧gαb000010.3110.3100.3Definition 10.
A mappingη∗:Sℤpk⟶Sℤppk−1is said to be anα-IF generalized gray map if for anyfα,gα∈Sℤpkwe have(37)η∗fα∧gαb=fα∧gαa,ifb=ηa;0,otherwise.Definition 11.
IfA¯is anα-IF code overℤpk, then it is said to be anα-IFℤpk-linear code if it is an image under theα-IF generalized gray map of a linear code overℤpk.Definition 12.
Consider anα-IF codeC; then, it is anα-IFℤpkcyclic code ifCis anα-IFℤpk-linear code and also if it is the image under theα-IF generalized gray map of a cyclic code overℤpk.Example 4.
Consider mappingsfα,gα:ℤ26⟶0,1where(38)fαx0,x1,x2,x3,x4,x5=1,ifx4=x5=0;0,otherwise,
and(39)gαx0,x1,x2,x3,x4,x5=0,ifx4=x5=0;1,otherwise.
Then,C=M,fα,gα is an α -IF linear code having length six over ℤ2 . Now, if we consider another code where fα,gα:ℤ43⟶0,1 defined asfα′x0,x1,x2=1,ifx2=0;0,otherwise, and gα′x0,x1,x2=0,ifx2=0;1,otherwise; then, C=M,fα,gα is an α -IF linear code having length 3 over ℤ4.Theorem 6.
Anα-IF gray mapη∗is a bijective map.Proof.
This is due to the fact thatη∗ is a one-to-one function.


## 3.4. Error Correction

The coding theory deals with encoding and decoding a message. During transmission of data, errors may occur; they can be detected and corrected by using different procedures such as parity check, syndrome decoding, and redundancy check which are for ordinary codes. Similarly, a method can be adopted in fuzzy codes for the confirmation about the codeword, that is, whether it belongs to the transmitted code or not. This can be done by considering the level set from which one can determine the confirmation degree as to whether a received codeword belongs to the original code. This method can be explained through the following example.Example 5.
Consider a moduleℤ24, where(40)ℤ24=0000,1000,0100,0010,0001,1100,1010,1001,0101,0110,0011,1011,1101,1110,0111,111.
LetA¯=M,fα,gα be an α-IF submodule where fα,gα:ℤ24⟶0,1 and(41)C=0000,0001,0010,1100,0011,1101,1110,1111,
which is a subset ofℤ24and also a linear code. Now, by transmitting this code, the received codeword has errors; assume that the received codewords are000;.01,010;0.001,011;0.01,100;0.1,101;0.01,110;\\0.1,001;0.01,111;0.999.
Table4 shows the values of fα,gα corresponding to the codewords of code C.
The level set is defined asA¯t1t2=a∈M:αofa≥t1,αoga≤t2; from this, we get for t1>0.9 and t2≤0.01 that the received codeword is in C.Table 4
Recovering of codewords
afαgαafαgαafαgαafαgα00001010000.90.101000.90.100100.990.0100010.990.0111000.990.0110100.90.110010.90.101010.90.101100.90.100110.990.0110110.90.111010.990.0111100.990.0101110.90.111110.990.01


## 3.5. Application

### 3.5.1. Sphere inα-IF Linear Code

Supposeℤpkm be a module and A¯=M,fα,gα be an α-IF submodule. If C is an α-IF linear code, then the level set Ct1t2 of C is also an α-IF linear code. As the elements in ℤpkm are the words of length m over the alphabet 0,1,2,…,pk−1 and the level set Ct1t2 also consists of the elements of ℤpkm, then for a member a∈Ct1t2 and for any integer r≥0, the sphere of radius r and center a is defined as follows.Theorem 7.
Consider a moduleM=ℤpkm; then, the sphere having radiusrwhere0≤r≤mcontains(42)m0+m1pk−1+m2pk−12+⋯+mrpk−1r,
words.Proof.
Consider a moduleℤpkm. Let x be a fixed word in ℤpkm. Then, the number of the words which differ from x in m position is mwpk−1w.Example 6.
LetM=ℤ22, whereℤ22=00,01,10,11. Here,k=1, p=2, m=2, and0≤r≤2. If we take a wordx=10, then its distances from the remaining codewords ared10,00=1, d10,01=2, d10,11=1, and the sum of these distances is equal to 4. Now,20+212−1+222−12=1+2+1=4, which is also equal to 4.


## 3.5.1. Sphere inα-IF Linear Code

Supposeℤpkm be a module and A¯=M,fα,gα be an α-IF submodule. If C is an α-IF linear code, then the level set Ct1t2 of C is also an α-IF linear code. As the elements in ℤpkm are the words of length m over the alphabet 0,1,2,…,pk−1 and the level set Ct1t2 also consists of the elements of ℤpkm, then for a member a∈Ct1t2 and for any integer r≥0, the sphere of radius r and center a is defined as follows.Theorem 7.
Consider a moduleM=ℤpkm; then, the sphere having radiusrwhere0≤r≤mcontains(42)m0+m1pk−1+m2pk−12+⋯+mrpk−1r,
words.Proof.
Consider a moduleℤpkm. Let x be a fixed word in ℤpkm. Then, the number of the words which differ from x in m position is mwpk−1w.Example 6.
LetM=ℤ22, whereℤ22=00,01,10,11. Here,k=1, p=2, m=2, and0≤r≤2. If we take a wordx=10, then its distances from the remaining codewords ared10,00=1, d10,01=2, d10,11=1, and the sum of these distances is equal to 4. Now,20+212−1+222−12=1+2+1=4, which is also equal to 4.


## 3.6. Genetic Code with Moduleℤ64

Amino acid sequence of a protein can be determined from the sequence or order of DNA and RNA molecules [26]. Although the information about protein sequence is found in the sequence of nucleotides in DNA, we cannot say that the proteins formed directly from the DNA. RNA molecules are also involved in the formation of proteins. RNA is composed of four bases which are adenine (A), guanine (G), cytosine (C), and uracil (U). One unit is formed from the three adjacent bases which are called a codon which codes for an amino acid. For instance, the amino acid methionine is encoded by the codon AUG. There are 64 codons; 20 amino acids are specified by the 61 codons that make up proteins. The remaining three codons do not code for any amino acid.There is another important aspect related to genetic code which is the algebraic structure of a genetic code from which some important information can be obtained [27]. Many scientists [28, 29] have studied the algebraic structure of codons. They explore various methods to model the genetic code mathematically and use binary representation for the four bases. Nemzer [30] established binary representation of genetic codes. Press et al. [31] described the HEDGES (Hash Encoded, Decoded by Greedy Exhaustive Search) error-correcting code to repair basic DNA errors. Rocha et al. [32] analyzed the DNA sequence generated by linear codes over the ring ℤ4. Bennenni et al. [33], Dinh et al. [34], and Gowthaman et al. [35] investigated DNA cyclic codes over rings.In the presented study, codon structure is investigated by involving the lattice structure of codons along with module over a ring. As there are 64 possible codons, we takeℤ64 module. From this, we conclude that α-IF submodule with module ℤ64 is a linear code. Consider a module M=ℤ64 and let fα,gα:ℤ64⟶0,1. Suppose f,g:ℤ64⟶L where L is a lattice consisting of 64 codons; the values for these functions and their composition are shown in Table 5. Here, A¯=M,fα,gα is an α-IF submodule so it is an α-IF linear code. We will use the following methodology.Table 5
Codons along with their respective degrees.
af(a)gaa(f(a))0UUUAAA11UCUAGA0.352UUGCAA0.43UCGAGC0.34GGUAAC0.45GCUCGA0.36UUCGAA0.357UCCAGG0.288GUGCAC0.359GCGCGC0.2810CUUAAG0.3511CCUGGA0.2812UUAUAA0.313UCAAGU0.2814GUCGAC0.315GCCCGG0.2516CUGCAG0.317CCGGGC0.2518AUUAAU0.319ACUUGA0.2520GUAUAC0.2821GCACGU0.222CUCGAG0.2823CCCGGG0.224AUGCAU0.2825ACGUGC0.226CUAUAG0.2527CCAGGU0.3528AUCGAU0.2529ACCUGG0.1530AUAUAU0.2531ACAUGU0.132UGUACA0.433UAUAUA0.334UGGACC0.3535UAGAUC0.2836GGUCCA0.3537GAUCUA0.2838UGCACG0.339UACAUG0.2540GGGCCC0.341GAGCUC0.2542CGUGCA0.343CAUGUA0.2544UGAACU0.2845UAAAUU0.2846GGCCCG0.2847GACCUG0.2848CGGGCC0.2849CAGGUC0.2850AGUUCA0.2851AAUUUA0.2852GGACCU0.2553GAACUU0.254CGCGCG0.2555CACGUG0.256AGGUCC0.2557AAGUUC0.258CGAGCU0.259CAAGUU0.1560AGCUCG0.261ACCUUG0.1562AGAUCU0.1563AAAUUU0Consider the latticeL comprises 64 codons and a module Z64=0,1,…,63 which also consists of 64 elements. Let a=0∈Z64; we know that f,g:Z64⟶L. Then, f0=UUU,g0=AAA, which are top and bottom elements of the lattice L as shown in [36]. As fα,gα:ℤ64⟶0,1, assign the values to these functions as fα0=1∈0,1, gα0=0∈0,1, where fα0+gα0≤1. Similarly, values can be assigned to the remaining codons, which is shown in Table 5. Then, by Definition 3, α-IF submodule with module Z64 is referred to as a linear code. By using Definition 4, we can compute the degrees of element t∈0,1; for example, for t=1, the degree is 1 and for t=0.3 the degree is 10. The remaining values can be computed in a similar way. Degrees of respective codons are also given in Table 5.The most important properties of amino acids are their hydrophobic and hydrophilic properties. As in lattice diagram, codons with second baseU show hydrophobic amino acids, and those with second base A show hydrophilic amino acids. There are total 16 codons that specify hydrophobic and hydrophilic amino acids, so we can define an α-IF linear code by considering the ℤ16 module. Let f,g:ℤ16⟶L and fα,gα:ℤ16⟶0,1, where lattice L consists of 16 elements. A comparison can be made between the two properties of amino acid by employing the concept of degree given in Definition 4. It can be seen in Table 6 that the average degree of hydrophobic amino acids is greater than the hydrophilic amino acids.Table 6
Codons withU or A as a second base which are coding for hydrophobic or hydrophilic amino acid.
afafαaDegreegagαaDegreeafafαaDegreegagαaDegree0UUU11AAA011AUA0.11AAC0.0122AUC0.122CAA0.0123CUA0.122AAG0.0534AUG0.23CAC0.0535CUC0.23GAA0.0536GUA0.23AAU0.147AUU0.254CAG0.148CUG0.254GAC0.149GUC0.254UAA0.1410UUA0.254CAU0.12211CUU0.33GAG0.12212GUG0.33UAC0.12213UUC0.33GAU0.2214GUU0.352UAG0.2215UUG0.352UAU0.251


## 4. Conclusion

Communication systems are designed for data transmission, working on the principle of encoding and decoding information. In classical coding theory, different procedures can be adopted to detect and correct errors that may arise during communication. The communication process is highly influenced by vagueness, inaccuracy, imprecision, and uncertainty. The classical coding methodologies are sometimes not efficient in handling such situations. Therefore, fuzzy logic is a viable option to handle such type of information. Lattice valued intuitionistic fuzzy sets are the generalization of basic fuzzy sets that incorporated the degree of both membership and nonmembership. Thus, they constitute a more efficient framework to model uncertain data. In this article, LIF-3 submodule andα-IF submodule are defined. These two structures are related by means of their level sets; that is, the level set of LIF-3 submodule is contained in the level set of α-IF submodule. After obtaining this result, further codes are discussed over α-IF submodule. Linear codes and cyclic codes are defined over α-IF submodule. Some important properties and results related to these codes are investigated. It is also concluded that this concept of α-IF linear codes is entirely applicable in genetic code. There are 64 codons that specify different amino acids, so Z64 module is considered here and α-IF linear codes are defined over Z64 module.In literature, conventional codes such as hamming codes and Hadamard codes are defined, which are constructed by considering vector spaces and fields. Modules are the extension of vector spaces. Thus, this work can be extended further by employing the present methodology and by using modules instead of vector spaces and fields. This can be done by investigating these codes overα-IF submodule by involving ℤpkn module over a ring ℤpk instead of the field F. Moreover, there are several generalizations of fuzzy sets like picture fuzzy sets [37], Pythagorean fuzzy sets [38], hesitant fuzzy sets [39], and neutrosophic sets [40] where the concept of lattices and codes can be extended to cope with the uncertainty in a better way. In “Application,” a link is made by incorporating the genetic code along with the concept of lattice valued intuitionistic fuzzy sets of type-3, and α-IF codes are defined by considering the modules Z64 and Z16. This methodology seems to be attractive for further investigation of genetic codes.



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*Source: 8148284-2022-06-23.xml*