On the Fuzzy Decomposition of Fuzzy Codewords with the Aid of Relative Weight

Accurate transmission of a message is always of prime importance in day-to-day life. Messages of paramount importance require greater caution while transmission, so that it is not disrupted by any means. But due to several problems there may be some sort of misinterpretation on the received message. Based on the above mentioned factors, in this paper, a new idea of the Unique Decipherability of Fuzzy Codewords, their decomposition and the ambiguity of fuzzy codewords based on their relative weight are proposed, which handles several special issues in Information Theory.


Introduction
The process of deciphering the correct message from a wrong one based on the parameters where it went wrong has always been the prime notion of studying coding theory. The notion of coding theory was first initially proposed by Shannon [4], in which he proposed the transmission of additional bits in a much smaller way. After Shannon, several studies [2,3,6,7] were carried out on studying the theories and varieties of codes, their decipherability. This was further developed by Hamming [9] who developed the ingenious method of detection of errors, and also several profound theories on correcting those errors. This theory of error correction comprises the study using Maximum Likelihood Decoding [5]. The notion of fuzzy sets was initially proposed by Zadeh [10]. The prime focus of fuzzy logic lies between assigning degree of association in the closed interval between 0 and 1, for all the members in a given collection. A new perspective of coding theory based on fuzzy logic was studied in [1]. Following the idea, the notions of Unique Decipherability of Fuzzy Code, it's fuzzy concatenative independence, fuzzy coding decomposition of fuzzy codewords are studied here along with their properties. The advantage of this method lies in the fact that every disrupted message has a unique membership value within the interval [0,1], thus helping us in identifying the proximity of the misinterpreted words. This way of partition is utilised in the process of Maximum Likelihood Decoding of Fuzzy Codewords studied in [8].
We have organized this paper as described below: Section -1 consists of Introduction and Preliminaries. In Section -2, the properties of Unique Decipherability of a fuzzy code, its fuzzy concatenative independence, fuzzy segregation and the notion of fuzzy coding partition of fuzzy codes are studied along with their properties.
Thus this weight is called a Maximum Relative Weight of codeword x of C . Definition 1.5. [1] The Relative Weight of a codeword x in (F 2 ) n is denoted by J(x) and it is defined by , J(y)}, for every x,y ∈ C Here, the members of the fuzzy code J(C) are called as fuzzy codewords.

Decomposition of a Fuzzy Codeword
This section is devoted to study the notion of Fuzzy Segregation of a fuzzy code, Unique Decipherability of Fuzzy Code, Fuzzy Concatenative Independence of Fuzzy Code, the fuzzy coding partition and the notion of total ambiguity of fuzzy code are defined and some of their properties are studied. Throughout this section C denotes a code comprising codewords of varying lengths. Definition 2.1. Let the collection of codewords, {C 1 , C 2 ,...,C n } be the segregation of C, such that each collection C i contains codewords of the same length. Let J(C i ) = {J(C i1 ), J(C i2 ),. . . ,J(C in )} be the respective relative weights associated with C. The collection J(C) = {J(C 1 ), J(C 2 ),...,J(C n )} is called the Fuzzy Segregation associated with C if each fuzzy codeword J(C ik )(k=1,2,...,n) ∈ J(C i ) has the same maximum relative weight.

Definition 2.2.
Let the collection of codewords, {C 1 , C 2 ,...,C n } be the segregation of C, such that J(C) is the fuzzy segregation associated with C. Then the fuzzy decomposition of an arbitrary fuzzy codeword J(C t ) is given by J(C t ) = J[⊕ C ij ] where i = 1,2,...,n and 1 ≤ j ≤ n.

Definition 2.3.
Let the collection of codewords, {C 1 , C 2 ,...,C n } be the segregation of C, such that J(C) is the fuzzy segregation associated with C. The collection J(C) = {J(C 1 ), J(C 2 ),...,J(C n )} is said to be a Uniquely Decipherable Fuzzy Code (denoted by UDFC), if there is a unique fuzzy decomposition of any arbitrary fuzzy codeword J(C it ) (i=1,2,...,n and t = 1,2,...,m). (i.e.,) If .., C in = C jm . A fuzzy code that is not a UDFC is an ambiguous fuzzy code. Definition 2.4. Let the collection of codewords, {C 1 , C 2 ,...,C n } be the segregation of C" such that J(C) is the fuzzy segregation associated with C. Let D = {D 11 , D 12 ,..., D 1m } (m = n), be another collection of codewords different from C such that J(D ik ) = J(C ij ) for any arbitrary fuzzy codeword J(C ij ) ∈ J(C i ) (i, j, k=1,2,...,n and i = j) The collection J(C) is said to be Fuzzy Concatenatively Independent if the following conditions holds: ..,n, j = 1,2,...,m and k = 1,2 ,..., n, j =k. (ii) Each fuzzy codeword in J(C) has a different relative weight. Definition 2.6. Let C be a code comprising codewords of varying lengths and let C = {C 1 , C 2 ,...,C n } be the segregation of C, such that J(C) is the fuzzy segregation associated with C. The collection J(C) is said to be a fuzzy coding decomposition if J(C) is fuzzy concatenatively independent and any arbitrary fuzzy codeword J(C t ) has a unique C p -Fuzzy Decomposition.
Remark 2.7. Let C be a code comprising codewords of varying lengths and let C = {C 1 , C 2 ,...,C n } be the segregation of C, such that J(C) is the fuzzy segregation associated with C. Suppose that an arbitrary fuzzy codeword J(C t ) has two fuzzy decompositions such that J(C t ) = J(C il ) = J(C jm ), where J(C il ) = J[C i1 ⊕ C i2 ⊕...⊕ C is ] and J(C jm ) = J[C j1 ⊕ C j2 ⊕ ... ⊕ C jt ] (i, j=1,2,...,n, i = j and s,t ≤ n). The fuzzy decomposition of J(C t ) is called a prime fuzzy decomposition if for l < s and m < t, J(C iv ) = J(C jw ) for any v = 1,2,...,l and w = 1,2,...,m. Remark 2.8. An arbitrary fuzzy codeword J(C t ), having two fuzzy decompositions such that .. ⊕ C jt ] (i = j and s,t ≤ n) can be uniquely represented as a prime fuzzy factorization J( Remark 2.9. Let C be a code comprising codewords of varying lengths and let C = {C 1 , C 2 ,...,C n } be the segregation of C, such that J(C) is the fuzzy segregation associated with C. Then for a unique fuzzy decomposition of an arbitrary fuzzy codeword J(C t ), there exists a unique C p -Fuzzy Decomposition of J(C t ). Thus for an arbitrary fuzzy codeword with two distinct C p -Fuzzy Decompositions, we have two distinct fuzzy decompositions. Proposition 2.10. Let C be a code comprising codewords of varying lengths and let C = {C 1 , C 2 ,...,C n } be the segregation of C, such that J(C) is the fuzzy segregation associated with C. The fuzzy segregation J( C) associated with C is a fuzzy coding decomposition iff for every prime fuzzy decomposition of an arbitrary fuzzy codeword J( .. ⊕ C jt ] an integer n exists, such that, for l ≤ s and m ≤ t, J(C il ), J(C jm ) ∈ J(C n ), i, j = 1,2,...,n and i = j, s,t ≤ n. Proof. Let us assume that J(C) is a fuzzy coding decomposition and let J( .. ⊕ C jt ] be the two prime fuzzy decompositions of an arbitrary fuzzy codeword J(C p ) and let J[C p1 ⊕ C p2 ⊕ ... ⊕ C pn ] be its unique C p -Fuzzy Decomposition. To prove our hypothesis we have to show that n = 1. On the contrary assume that n > 1. Since, J(C) is a fuzzy coding decomposition it has a unique C p -Fuzzy Decomposition and hence for l < s and m < t, s,t ≤ n we have J( .. ⊕ C jm ], i = j, i, j = 1,2,...,n which is not possible since J(C p ) is a prime fuzzy decomposition. Hence n = 1.
To show that J(C) is a fuzzy coding decomposition we initially prove that the members of J(C) are fuzzy concatenatively independent. Let us assume the contrary that they are not fuzzy concatenatively independent. Then by Definition 2.4, for any two fuzzy codewords J(C il ) ∈ J(C i ) and J(C jm ) ∈ J(C j ), where 1 ≤ l ≤ n and 1 ≤ m ≤ n and i, j = 1,2,...,n, i = j, J(C il ) ..,s) and J(C jn ) ∈ J(C j ) (n = 1,2,...,t). From Remark 2.8, J(C p ) has a prime fuzzy decomposition such that J( , which is a contradiction as C im ∈ C i (m = 1,2,...,s) and C jn ∈ C j (n = 1,2,...,t) whenever i = j.
To complete the proof we now show that, any arbitrary fuzzy codeword has a unique J(C p ) fuzzy decomposition. Suppose on the contrary if there are two distinct J(C p ) fuzzy decompositions , then by Remark 2.9, we have two distinct fuzzy decompositions forJ(C p ) , Since i =j, we have by Remark 2.8, a prime fuzzy decomposition such that Also, by the previous part of the proposition, the fuzzy code J(C) is fuzzy concatenatively independent and hence, J(C i1 ) = J(C j1 ) (i, j = 1,2,...,n and i = j). Let N denote the number of vectors in an arbitrary fuzzy codeword and suppose that N [(C j1 )] < N [(C i1 )], then it is clear that N [(C j1 )] ≥ 2. Since J(C) has a C p fuzzy decomposition, for C j ∈ C if J(C j1 ) ∈ J(C j ) and J(C j2 ) / ∈ J(C j ).
Thus an arbitrary fuzzy codeword in a prime fuzzy decomposition J(C h ) (1 ≤ h ≤ n) cannot occur in both decompositions of J(C j1 ) and J(C j2 ). Hence, we have Thus J(C 1 ) is expressed both in terms of J(C j1 ) and J(C i1 ). But since they are fuzzy concatenatively independent, we have J(C 1 ) ≤ {J(C i )} (i=1,2,...,n).
Let J(C j1 ) = J[C 1 ⊕ C 2 ⊕ ... ⊕ C m ], 1 ≤ m < n. By the hypothesis, n > 1 and N [(C j1 )] ≥ 2, therefore a prime fuzzy factor J(C m+1 ) ≯ J(C 1 ) or J(C m+1 ) ≯ J(C j1+1 ), which implies, J(C m+1 ) ≤ J(C 1 ) and J(C m+1 ) ≤ J(C j1+1 ). But J(C 1 ) ≤ J(C i ) and J(C j1+1 ) ≤ J(C j ) and also J(C m+1 ) has a prime fuzzy decomposition, which is a contradiction. Thus the fuzzy segregation J(C) has a unique J(C p ) fuzzy decomposition and hence J(C) is a fuzzy coding decomposition. Definition 2.11. Let the collection of codewords, {C 1 , C 2 ,...,C n } be the segregation of C, such that J(C) is the fuzzy segregation associated with C. The fuzzy segregation J(C) is a trivial fuzzy segregation if J(C) has only one fuzzy segregation. Definition 2.12. The fuzzy segregation J(C) is called a discrete fuzzy segregation if each fuzzy segregation in J(C) has only one fuzzy codeword.