A Novel Way of Detecting and Correcting Transmitted Errors using Fuzzy Logic

Transmission and receival of messages always play an important role on various channels. Besides the transmission it is also necessary to be sure that only the exactly transmitted is transferred and the original message is not trampled due to the presence of some disturbance sources in the channel. Inspite of precautions if there is an error in transmission it is mandatory to recover the original message with the aid of several decoding measures. One among the decoding measures is the method of implementing the notion of fuzzy logic, which has its wide applications in several fields. Thus in this paper with the help of the concept of relative weights, the codes are studied in the sense of fuzzy logic to decode the transmitted message, detect errors in transmission and to correct them. Hence the technique of detecting and correcting errors presented in this paper provides a better outcome in comparison with several existing such methods. The prime aim of studying these concepts in fuzzy logic, is it’s precision and accuracy in terms of membership values which avoids the disparities that arises in other methods.


Introduction
Message disruption has always been a cause of concern on various transmission channels. Disruption results in errors which makes the receiver to obtain a totally different message sent by the sender. The study of such disruptions and their corrections in transmission is the sole purpose of coding theory. The notion of coding theory was first initially proposed by [2], in which he proposed the transmission of additional bits in a much smaller way. This was further developed by [6] who developed the ingenious method of error detecting and error correcting codes [5,[7][8][9][10]. The notion of fuzzy sets was initially proposed in [11]. A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. These membership grades are represented by real number values ranging in the closed interval between 0 and 1 [3] . A new perspective of coding theory based on fuzzy sets was studied in [1]. In this paper, the notions of Fuzzy Hamming distance of Fuzzy Codewords, Fuzzy Hamming Weight of Fuzzy Codewords, Linear Fuzzy codewords and Equivalent Fuzzy codewords are studied along with some of their properties .

Preliminaries
The preliminaries required for the study of this paper are as follows: Definition 2.1: [4] The Exclusive Or is a basic computer operation denoted by XOR or ⊕, which takes two individual bits ∈ {0,1} and ′ ∈ {0,1} and yields , J(y)}, for all x,y ∈ C Here, the members of the fuzzy code J( ) are called as fuzzy codewords.

Fuzzy Linear Codes
In this section, the concepts of Fuzzy Hamming Weight, Fuzzy Hamming Distance for fuzzy codewords, Lower Bound and Linearity of a Fuzzy code and the decoding and error correction of fuzzy codewords are studied with suitable examples along with some of their properties.

Definition 3.3
The Fuzzy Hamming Distance between two codewords C and C , (i ≠ j, 1 ≤ i, j ≤ n) in ℭ (denoted by ℱℋ (C , C ))) is a function from {C 1 ,C 2 ,...,C } → I = [0, 1] and it is defined as the relative weight of the number of places of the vectors by which the two codewords differ.
Suppose for any two codewords C and C , (l ≠ k, 1 ≤ l, k ≤ n) in ℭ, the number of vectors is 1, then in this case the Fuzzy Hamming Distance is considered for the vectors which is denoted by ℱℋ (x 1 , x 1 ), where x , x are single vectors of C and C respectively and it is defined as . Now, the number of places on which the vectors differ are 2, and the maximum relative weight of codewords in ℭ is 15. Hence, the Fuzzy Hamming Distance is 2/15.
Proof. By Definition 3.3, are the vectors in C and C respectively, and For n = 1, if C ≠ C , then ℱℋ (C , C ) = 1 and we have either We now prove for n > 1, by the previous case, for each vector in the codeword we have, , where x , x , x (i,j,k = 1,2,...,n) are the vectors of the codewords C , C and C (i,j,k = 1,2,...,n) of ℭ respectively. Thus, which is the triangle inequality.

Remark 3.6 For any two codewords
Proof. For any two codewords C and C , (i≠ j, i, j = 1,2,...,n) in ℭ, let 'x' denote the number of places of the vectors by which the codewords C and C differ and let the maximum relative weight of the codewords in ℭ be d. Hence, by Definition 3.3, the Fuzzy Hamming Distance between C and C is x/d. Now, by Definition 2, C ⊕ C gives '1' only at places where the vectors of the codewords differ. Hence, for the codewords in ℭ with maximum relative weight d, the Fuzzy Hamming Weight of

Maximum Likelihood Decoding of Fuzzy Codewords
It is not always possible for the receiver to exactly receive the original message. When a message is sent, some disturbances can alter the original message to some other message. Whenever the original message is not received by the receiver, we say that there is an in the message. Due to the presence of errors, the original codeword, that is sent gets changed and a new codeword reaches the receiver. This new codeword is called the . This transmitted codeword will not give the original message that was intended to be sent, which results in miscommunication thus leading to adverse effects. Hence, it is necessary to identify the error in transmission and rectify the mistake. The principle of Maximum Likelihood Decoding in Fuzzy Codes compares the Fuzzy Hamming Distance of the transmitted fuzzy codeword with the original fuzzy codewords. Among this, the original fuzzy codeword that gives the Lower Bound of the fuzzy code with the transmitted fuzzy codeword is identified to be the error rectified fuzzy codeword. Thus, we have the following Definition 3.10.

Fuzzy Erasures
Another important parameter, which is taken into account while decoding the original transmitted fuzzy codewords is the fuzzy erasure. Presence of a fuzzy erasure signifies the loss of relative weight of a particular bit or several bits. In other words, the relative weights are not altered due to the change in codewords, rather it goes missing.

Definition 3.12
Let the fuzzy codewords be transmitted to the receiver and during the transmission owing to the disturbances some relative weights of the fuzzy codewords are erased and a blank space is received by the receiver instead of the relative weights. These blank spaces are called the Fuzzy Erasures.  (1) ⇒ (4) Let the transmitted codeword be , which has some vectors missing on it. Then the realtive weight of this transmitted codeword is, J( ) − x, where 'x' denotes the relative weight of the missing bits.
Let w , v , u (1 ≤ i, k, l ≤ n) denote the vectors in the codewords , and respectively. We claim that there exists a unique codeword such that for the vectors J(w ) = J(v ) (i, k = 1,2...,n), at every place except at the erasure part. Here J(w ) and J(v ) denote the relative weight of the vectors on the codeword and respectively. Assume the contrary that it is not unique. Then there exists another distinct codeword such that J(u ) = J(v ), at every places except at the erasure part. This implies that J(w ) = J(u ), at every place except at the erasure part. Let (x) denote the number of fuzzy codewords in the fuzzy erasure part . Then ℱℋ ( , ) ≤ (x) ≤ ℒℬ(J(ℭ)) − 1

Construction of Extended Fuzzy Code for length 3
Suppose that J(ℭ) is a linear fuzzy code. Let a codeword in ℭ (i = 1,2,...,n) be indexed with vectors as c 1 c 2 c 3 . Now a vector is added to such that, where w(c) is the number of non-zero entries of each codeword. Now by Proposition 3.17, since J(ℭ) is a linear fuzzy code, the minimum fuzzy hamming weight of ℒℬ( ( )) will be equal to ℒℬ( ( )) and hence we have