( Amudhambigai & Neeraja 2019) F n is the collection of all ordered n − tuples

p = p 1 , p 2 , … , p k , Where each p _i in F q . The elements of F n are referred to as words or vectors.

Ozkan & Mehmet (2002) Let us assume that C has a as a code word. The relative weight of a code word a , denoted as

R w ( a ) , is the sum of p 1 , p 2 , … , p k if p 1 , p 2 , … , p k are defined as the positions of 1 _s in a . For instance, R w ( a ) = 1 + 3 + 4 + 5 = 13 if a = 10111 is a code word of C in F 5 .

1 + 2 + 3 + … + n is the relative weight of 111…1 , which is a code word of C in F 2 n . This can be articulated as the sum of the first n positive integers. The largest relative weight of code C in F 2 n is referred to as this matter. Where n is the length of a code word, where R wr is a mapping defined as R wr : C → [ 0 , 1 ] for C ∈ F ₂ ⁿ and R wr ( a ) ∈ [ 0 , 1 ] , for any a ∈ C .

( Ozkan & Mehmet 2002) Let { c 1 , c 2 , … , c n } is the code word of C with length n. Let C be a code.

For each i = 1 , 2 , … , n , let R wr ( c i ) be a fuzzy code word related to the code words c i , c j in C . Then the fuzzy code intersection and union of any two fuzzy codes R wr ( c i ) and R wr ( c j ) , respectively, are given as follows: (i)

R wr ( c i ) ⋂ R wr ( c j ) = min { R wr ( c i ) , R wr ( c j ) }

( Amudhambigai and Neeraja 2019) For any fuzzy code R wr ( c i ) , the complement is c ( R wr ( c i ) ) is calculated by subtracting from 1 the relative weight of each member of C .

That is, c ( R wr ( c i ) ) = 1 − R wr ( c i ) (1)

( Molodtsov 2011) Consider the idea of a set of parameters represented by E and U be the notion of a starting universe of items. P ( U ) is the power set of U and A ⊆ E . A pair ( F , A ) is a soft set over U if and only if F is a mapping given by F : A → P ( U )

In other words, the soft set is a parameterized family of subjects from the set U.

Consider the set of e ∈ A, or the set of e − approximate components of the soft set, where F ( e ) can be arbitrary, some of which can have nonempty intersection, and some of which can be empty. This applies to every F ( e) from this family for e in A . where the attributes, traits, or properties of objects are usually the arguments. We might consider Zadeh’s fuzzy set as a subset of the soft set. Given a fuzzy set A , its membership function is μ A , that is, μ _A is a mapping from U into [ 0 , 1 ] , that is μ A : U → [ 0 , 1 ]

( Majumdar & Samanta, 2010) The set of all fuzzy subsets within a universal set U is denoted by I U .

Assume A ⊆ E and that E is a parameter set. Then, a pair ( F , A ) is a fuzzy soft set over U , where F is a mapping denoted by F : A → I U

( Ali et al. 2018) Assume that a vector space of size n over the field K is represented by W = K n . ( F , E ) is as of t algebraic linear code over K . For every F ( e i ) ; 1 ≤ i ≤ t , the symbol F ( E ) = F ( e 1 ) , … , F ( et ) in x indicates a linear algebraic code of W . The number of linearly independent elements of F ( e i ) is denoted by d i , which is the dimension of F ( e i ) . Each F ( e i ) ∈ F ( E ) in this instance represents a linear algebraic code. The soft dimension of ( F , E ) is ( F , E ) = { d 1 , d 2 … , d t } and the number of soft code words of ( F , E ) is given by: n ( F , E ) = | F ( e 1 ) | × | F ( e 2 ) | × … × | F ( e t ) | (2)

Where 1 ≤ i ≤ t .

Keep in mind that dim ( F , A ) represents the soft dimension ( F , E ) .

( Majumdar & Samanta 2010). The universal set of elements is = { x 1 , x 2 , … , x n } , and the universal set of parameters is E = { e 1 , e 2 , … , e m } . We shall refer to the pair ( U , E ) as a soft universe. Let μ be a fuzzy subset of E , that is, μ : E → [ 0 , 1 ] , and let F : E → I U . where the set of all fuzzy subsets of U is represented by I U . Assume that F μ represents the mapping. Let a function Fμ such that

Fμ : E → I U × I have the following definition:

Fμ ( e ) = F ( e ) , μ ( e ) ) , where ( e ) ∈ I U .

A generalised fuzzy soft set (in short GFSS) over the soft universe ( U , E ) is then denoted by Fμ . Here, for every parameter e i , Fμ ( e i ) = ( F ( e i ) , μ ( e i ) ) showed both the degree of belongingness that μ ( e i ) represents and the degree of belongingness of the elements of U in F ( e i ) .

( Majumdar & Samanta 2010) Let Fμ and G σ be two generalised fuzzy soft sets over ( U , E ) . Now Fμ is said to be a generalised fuzzy soft subset of G σ if (i)

μ ⊆ σ

( Majumdar and Samanta 2010) Consider a generalised fuzzy soft set over ( U , E ) , denoted by Fμ . Then, F ∁ represents the complement of Fμ , which is defined as F ∁ = G σ , where G ( e ) = F ∁ and σ ( e ) = μ ∁ ( e ) , for all e ∈ E .

( Majumdar & Samanta 2010) Fμ ⋃ G σ represents the union of two generalised fuzzy soft sets Fμ and G σ . This results in a generalised fuzzy soft set H ν , which is defined as H ν : E → I U × I such that H ν ( e ) = ( H ( e ) , ν ( e ) ) where H ( e ) = F ( e ) ° G ( e ) , ν ( e ) = μ ( e ) ° σ ( e ) , and ° denotes a t-co norm.

( Majumdar & Samanta 2010) Fμ ⋃ G σ represents the intersection of two generalised fuzzy soft sets, Fμ and G σ . This leads to a generalised fuzzy soft set H ν , which is defined as

H ν : E → I U × I such that H ν ( e ) = ( H ( e ) , ν ( e ) ) , where ∗ is a t-co norm and H ( e ) = F ( e ) ∗ G ( e ) and ν ( e ) = μ ( e ) ∗ σ ( e ) .

( Majumdar & Samanta 2010) ∅ θ : E → I U × I indicates a generalised null fuzzy soft set, which is referred to as a generalised fuzzy soft set, such that ∅ θ ( e ) = ( F ( e ) , θ ( e ) ) , where ∅ ( e ) = 0 ¯ (null set or no element is included under any parameter within the soft set), and, θ ( e ) = 0 , for all e ∈ E .

( Majumdar & Samanta 2010) A generalised absolute fuzzy soft set, represented by

A α : E → I U × I , is refered to as a generalised fuzzy soft set, where

A α ( e ) = F ( e ) , α ( e ) is defined by A ( e ) = 1 ¯ (maximal soft set or where every parameter is associated with the whole set under consideration) and α ( e ) = 1 , for all e ∈ E .

Consider C ⊆ F 2 4 parameter set E, and define a fuzzy set on this parameter set μ( e) ∈ [0,1] for any set of R _wr ( a). And let R wr ( a ) = { R wr ( 0100 ) , R wr ( 1010 ) , R wr ( 1011 ) , R wr ( 1110 ) , R wr ( 0001 ) , R wr ( 0111 ) , R wr ( 1100 ) , R wr ( 1111 ) }

= { 0.2 , 0.5 , 0.8 , 0.6 , 0.4 , 0.9 , 0.3 , 1 } be a fuzzy code of the eight computers under consideration.

Let E = { e 1 , e 2 , e 3 , e 4 } be the set of qualities of the given computers, where the symbols e 1 denote a super computer, e 2 a micro computer, e 3 a workstation computer, and e 4 a personal computer.

Let μ : E → [ 0 , 1 ] be a fuzzy code subset of R _wr : C → [0,1] defined as follows: μ ( e 1 ) = R wr ( 0110 ) = 0.5 ; μ ( e 2 ) = R wr ( 1001 ) = 0.6 ; μ ( e 3 ) = R wr ( 1100 ) = 0.3 ; and μ ( e 4 ) = R wr ( 1011 ) = 0.8 ;

And we define a function

Fμ : E → P ( R wr ( a i ) ) × R wr ( a i ) , be defined as follows: F μ ( e 1 ) = ( { R wr ( 0001 ) 0.4 , R wr ( 0111 ) 0.9 , R wr ( 1110 ) 0.6 , R wr ( 1010 ) 0.4 } , 0.5 ) F μ ( e 2 ) = ( { R wr ( 1011 ) 0.8 , R wr ( 1110 ) 0.6 , R wr ( 1010 ) 0.4 , R wr ( 1111 ) 1 } , 0.6 ) F μ ( e 3 ) = ( { R wr ( 1110 ) 0.6 , R wr ( 0001 ) 0.4 , R wr ( 1011 ) 0.8 , R wr ( 0111 ) 0.9 } , 0.3 ) F μ ( e 4 ) = ( { R wr ( 0101 ) 0.6 , R wr ( 1100 ) 0.3 , R wr ( 1010 ) 0.4 , R wr ( 1111 ) 1 } , 0.8 )

A Generalised fuzzy soft code over ( R wr ( a ) , E ) is then F μ .

This can be represented in matrix form as follows: F μ = [ 0.4 0.9 0.6 0.4 | 0.5 0.8 0.6 0.4 1 | 0.6 0.6 0.4 0.8 0.9 | 0.3 0.6 0.3 0.4 1 | 0.8 ] where the last column represents the values of μ ( e i ) , the row vectors i th represent F μ ( e i ) , and the column vector i th represents R wr ( a i ) for some a i ∈ F 2 4 . This will be referred to as the membership matrix of F μ .

The GFSC is then can be defined as

F μ ( e i ) , ( e i ) = ( F ( e i ) , μ ( e i ) ) , ∀ e ∈ E , F ( e ) ∈ P ( R wr ( a i ) ) and ∀ i .

Given two GFSCs over ( R wr ( a ) , E ) , let F μ and G σ . Specifically, F μ is a subset of G σ that is considered a generalised fuzzy soft code if i

μ ⊆ σ

Here, we compose F μ ⊆ G σ .

The GFSC F μ over ( R wr ( a ) , E ) is the one presented in example 1. Let G σ be an additional GFSC defined as follows over ( R wr ( a ) , E ) . G σ ( e 1 ) = ( { R wr ( 1001 ) 0.5 , R wr ( 0111 ) 0.9 , R wr ( 1011 ) 0.8 , R wr ( 0110 ) 0.5 } , 0.6 ) G σ ( e 2 ) = ( { R wr ( 1011 ) 0.8 , R wr ( 0110 ) 0.5 , R wr ( 1110 ) 0.6 , R wr ( 1111 ) 1 } , 0.8 ) G σ ( e 3 ) = ( { R wr ( 1011 ) 0.8 , R wr ( 1010 ) 0.4 , R wr ( 0111 ) 0.9 , R wr ( 1111 ) 1 } , 0.5 ) G σ ( e 4 ) = ( { R wr ( 1011 ) 0.8 , R wr ( 1010 ) 0.4 , R wr ( 1001 ) 0.5 , R wr ( 1111 ) 1 } , 0.7 )

Here σ and R wr ( a ) are the respective mapping given by σ : E → [ 0 , 1 ] , and R wr ( a ) : C → 0 , 1 ]

Then, a generalised fuzzy soft code subset of G σ is F μ .

Matrix representation of G σ is given by G σ = [ 0.5 0.9 0.8 0.9 | 0.6 0.8 0.7 0.6 1 | 0.8 0.8 0.4 0.9 1 | 0.5 0.8 0.4 0.5 1 | 0.7 ]

(Intersection of GFSC). F μ ⋂ G σ represents the GFSC intersection of two GFSCs of F μ and G σ over ( R wr ( a ) , E ) , and is defined as Fμ ⋂ G σ = min { ( F μ ( e ) , μ ( e ) ) , ( G σ ( e ) , σ ( e ) ) , ∀ e ∈ E } . (14)

(Union of GFSC) F μ ⋃ G σ represents the GFSC union of two GFSCs of F μ and G σ over a soft universe ( R wr ( a ) , E ) , and is defined as Fμ ⋃ G σ = max { ( F μ ( e ) , μ ( e ) ) , ( G σ ( e ) , σ ( e ) ) , ∀ e ∈ E } . (15)

(Complement of GFSC). The following notation represents the complement of the GFSC of F μ : ( F μ ) ∁ or Subtracting the GFSC of each member of ( F μ ( e ) , μ ( e ) ) from 1 yield F ∁ μ ∁ . That is, F ∁ μ ∁ = ( 1 − F μ ( e ) , 1 − μ ( e ) (16)

If F μ and G σ represent any GFSC over a shared soft universe ( R wr ( a ) , E ) , then the following result is hold: (i)

F μ = F μ ⋂ F μ

The proof follows directly from the definition.

The following properties are hold, if ( F , A ) and ( G , B ) are two generalised fuzzy soft codes over the common universe R wr ( a ) . (i)

( ( F , A ) ⋃ ( G , B ) ) ∁ = ( F , A ) ∁ ⋂ ( G , B ) ∁

(i)

( ( F , A ) ⋃ ( G , B ) ) ∁ = ( F ∁ , ¬ A ) ⋂ ( G ∁ , ¬ B ) = ( H ∁ , ¬ A × ¬ B ) ,

where H ( ¬ e , ¬ e ′ ) = F ∁ ( e ) ⋃ G ∁ ( e ′ ) = ( H , ¬ ( A × B ) ) ) ⇒ ( F , A ) ∁ ⋃ ( G , B ) ∁ = ( H , ¬ ( A × B ) )

Assume that, ( ( F , A ) ⋂ ( G , B ) ) = ( H , A × B ) ) ⇒ ( ( F , A ) ⋂ ( G , B ) ) ∁ = ( H , A × B ) ∁ = ( Hc , ¬ ( A × B ) ) , ∀ ( e , e ′ ) ∈ A × B ) ⇒ H ∁ ( ¬ e , ¬ e ′ ) = ( H ( e , e ′ ) ) ∁ = ( F ( e ) ⋂ G ( e ′ ) ) ∁ = ( F ( e ) ) ∁ ⋃ ( G ( e ′ ) ) ∁ = F ∁ ( ¬ e ) ⋃ Gc ( ¬ e ′ ) = ( H , ¬ ( A × B ) ) ) ⇒ ( ( F , A ) ⋂ ( G , B ) ) ∁

Thus, from our discussion we get the result, ( ( F , A ) ⋃ ( G , B ) ) ∁ = ( F , A ) ∁ ⋂ ( G , B ) ∁

The proof of (ii) is similar to the above.

A generalised null fuzzy soft code, represented by F μ , is considered to be a generalised null fuzzy soft code if

F μ : E → P ( R wr ( a ) ) × R wr ( a ) and F μ ( e ) = ( F ( e ) , μ ( e ) ) , where F ( e ) = 0 ¯ and μ ( e ) = 0 , ∀ e ∈ E .

A GFSC is said to be a generalised absolute fuzzy soft code, denoted by G σ if G σ : E → P ( R wr ( a ) ) × R wr ( a ) , where G σ ( e ) = ( G ( e ) , σ ( e ) ) is defined by G ( e ) = 1 ¯ , ∀ e ∈ E , and σ ( e ) = 1 ∀ e ∈ E .

Given any GFSC over ( R wr ( a ) , E ) , let’s say F μ . The generalised null fuzzy soft code and generalised absolute fuzzy soft code over ( R wr ( a ) , E ) denoted as G σ and H θ , respectively. Then, the following result is hold: (i)

F μ ⋂ G σ = G σ

(i) Let F μ be any GFSC over ( R wr ( a ) , E ) and let G σ be the generalised null fuzzy soft code over ( R wr ( a ) , E ) . Then F μ ⋂ G σ = F μ ( e ) ⋂ G σ ( e ) = ( F ( e ) , μ ( e ) ) ⋂ ( G ( e ) , σ ( e ) ) = ( F ( e ) , μ ( e ) ) ⋂ ( 0 ¯ , 0 ) = ( 0 ¯ , 0 ) = G σ

(ii-v) Similar to the proof of (i)

Given a universal set U = R wr ( a ) , consider two generalised fuzzy soft codes over R wr ( a ) : ( F , A ) and ( G , B ) . Then, the addition operation modulo between ( F , A ) and ( G , B ) is represented by ( F , A ) ⊕ ( G , B ) , and the multiplication operation modulo between ( F , A ) and ( G , B ) is represented by ( F , A ) ⊙ ( G , B ) . The interpretation of these notations are as follows: (i)

( F , A ) ⊕ ( G , B ) = ( H , a × b ) , where ( H , a × b ) = F ( a ) ⋃ G ( b )