Comparative Study of Generalized Quantitative-Qualitative

Research ArticleComparative Study of GeneralizedQuantitative-Qualitative Inaccuracy Fuzzy Measures forNoiseless Coding Theorem and 1:1 Codes

H. D. Arora1 and Anjali Dhiman2

1Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Noida 201301, India2Department of Mathematics, Ideal Institute of Management and Technology, GGSIP University, Delhi 110092, India

Correspondence should be addressed to Anjali Dhiman; [email protected]

Received 3 March 2015; Accepted 1 April 2015

Academic Editor: Ram U. Verma

Copyright © 2015 H. D. Arora and A. Dhiman.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

In coding theory, we study various properties of codes for application in data compression, cryptography, error correction, andnetwork coding. The study of codes is introduced in Information Theory, electrical engineering, mathematics, and computersciences for the transmission of data through reliable and efficient methods. We have to consider how coding of messages canbe done efficiently so that the maximum number of messages can be sent over a noiseless channel in a given time. Thus, theminimum value of mean codeword length subject to a given constraint on codeword lengths has to be founded. In this paper,we have introduced mean codeword length of order 𝛼 and type 𝛽 for 1:1 codes and analyzed the relationship between averagecodeword length and fuzzy information measures for binary 1:1 codes. Further, noiseless coding theorem associated with fuzzyinformation measure has been established.

1. Introduction

In the 1940s, a new branch of mathematics was introducedas Information Theory. Information Theory considered theproblems of how to process information, how to store infor-mation, how to retrieve information, and hence decision-making. Information Theory deals with the study that howinformation can be transmitted over communication chan-nels. In 1924 and 1928, Nyquist [1, 2] first studied the infor-mation measures and Hartley [3] studied that informationmeasureswere logarithmic in nature.Newproperties of infor-mation sources and communication channels were publishedby Shannon [4] in his research paper “AMathematicalTheoryof Communication.” Wiener [5] also started consideringresults similar to Shannon. A large number of applications ofInformation Theory in various fields of social, physical, andbiological sciences have been developed, for example, eco-nomics, statistics, accounting, language, psychology, ecology,pattern recognition, computer sciences, and fuzzy sets. Later,Renyi [6] introduced entropy of order 𝑟, which is the limitingcase of Shannon entropy.

Fuzzy set theory was introduced by Zadeh [7]. It relates tothe uncertainty occurring in the human cognitive processes.Fuzzy set theory has found applications in various engi-neering, biological sciences, and business. Zadeh introducedfuzzy entropy, a measure of fuzzy information based onShannon’s entropy.

Let 𝐴 be a fuzzy set in the universe of discourse Ω. Wedefine membership function as 𝜇

𝐴: Ω → [0, 1], where

𝜇𝐴(𝑥) is a degree of membership for each 𝑥 ∈ Ω.The following properties were given by De Luca and

Termini [8] which are to be satisfied by fuzzy entropy.(1) Fuzzy entropy is the minimum iff set is crisp.(2) Fuzzy entropy is the maximum when membership

value is 0.5.(3) Fuzzy entropy decreases if set is sharpened.(4) Fuzzy entropy of a set is the same as its complement.

In coding theory, we study various properties of codes forapplication in data compression, cryptography, error correc-tion, and network coding. The study of codes is introduced

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2015, Article ID 258675, 6 pageshttp://dx.doi.org/10.1155/2015/258675

2 International Journal of Mathematics and Mathematical Sciences

in Information Theory, electrical engineering, mathematics,and computer sciences for the transmission of data throughreliable and efficient methods. We have to consider how cod-ing of messages can be done efficiently so that the maximumnumber of messages can be sent over a noiseless channel ina given time. Thus, the minimum value of mean codewordlength subject to a given constraint on codeword lengths hasto be founded. Let 𝑋 be random variable with each value𝑥𝑖, 𝑖 = 1, 2, . . . , 𝑛, that generates messages represented by

the set {𝑎1, 𝑎2, . . . , 𝑎

𝐷} which need to be transmitted. This

set is called code alphabet and sequence assigned to each𝑥𝑖is called codeword. The codeword length denoted by 𝑛

𝑖

associated with 𝑥𝑖satisfying Kraft’s inequality is given by

𝐷−𝑙1+𝐷−𝑙2+ ⋅ ⋅ ⋅ +𝐷

−𝑙𝑛

≤ 1, (1)

where 𝐷 is the size of alphabet. Further, codes are chosen tominimize average codeword length which is given by

𝐿 =

𝑛

∑

𝑖=!

𝑛𝑖𝑝𝑖, (2)

where𝑝𝑖is the probability of the occurrence of 𝑥

𝑖. Now, Shan-

non’s [4] noiseless coding theorem for uniquely decipherablecodes is given by

𝐻(𝑃)

log𝐷≤ 𝐿 <

𝐻 (𝑃)

log𝐷+ 1 (3)

finding the lower bounds on 𝐿 in terms of Shannon’s entropy𝐻(𝑃).

Campbell [9] proposed a special exponentiated meancodeword length of order 𝛼 for uniquely decipherable codesgiven by

𝐿𝛼=

𝛼

1 − 𝛼log𝐷[

𝑛

∑

𝑖=1𝑝𝑖𝐷(1−𝛼)𝑛

𝑖/𝛼] (4)

and proved a noiseless coding theorem. Campbell analyzedthat its lower bound lies between 𝑅

𝛼(𝑃) and 𝑅

𝛼(𝑃)+ 1, where

𝑅𝛼(𝑃) = (1 − 𝛼)−1log

𝐷[∑𝑛

𝑖=1 𝑝𝛼

𝑖]; 𝛼 > 0, 𝛼 = 1.

The above is Renyi’s measure of entropy of order 𝛼. As𝛼 → 1, it is easily shown that 𝐿

𝛼→ 𝐿 and𝑅

𝛼(𝑃) approaches

𝐻(𝑃).For uniquely decipherable code, a weighted average

length was given by Guiasu and Picard [10]:

𝐿 =

𝑛

∑

𝑖=1

𝑢𝑖𝑛𝑖𝑝𝑖

∑𝑛

𝑖=1 𝑢𝑖𝑝𝑖. (5)

This is defined as the average cost of transmitting letters 𝑥𝑖

with probability 𝑝𝑖and utility 𝑢

𝑖.

A quantitative-qualitativemeasure of entropywas definedby Belis and Guiasu [11] for which Longo [12] gave the lowerbound for useful mean codeword length. Further, a noiselesscoding theorem was verified by Guiasu and Picard by intro-ducing lower bound for useful mean codeword length. AndGurdial and Pessoa [13] proved noiseless coding theorem bygiving lower bounds for useful mean codeword lengths of

order 𝛼 in terms of usefulmeasures of information of order 𝛼.The following information measure was introduced by Belisand Guiasu:

𝐻(𝑃,𝑈) = −

𝑛

∑

𝑖=1𝑢𝑖𝑝𝑖log𝑝𝑖. (6)

Similarly, quantitative-qualitative information measure wasgiven by Taneja and Tuteja:

𝐻(

𝑃

𝑄

,𝑈) =

𝑛

∑

𝑖=1𝑝𝑖𝑢𝑖log(

𝑝𝑖

𝑞𝑖

) . (7)

Further, Bhaker and Hooda gave the following informationmeasures:

𝐻(

𝑃

𝑄

,𝑈) =

∑𝑛

𝑖=1𝑢𝑖𝑝𝑖log (𝑝

𝑖/𝑞𝑖)

∑𝑛

𝑖=1𝑢𝑖𝑝𝑖

,

𝐻𝛼(

𝑃

𝑄

,𝑈) =

11 − 𝛼

∑𝑛

𝑖=1𝑢𝑖𝑝𝛼

𝑖𝑞1−𝛼𝑖

∑𝑛

𝑖=1 𝑢𝑖𝑝𝑖, 𝛼 = 1.

(8)

Now, Baig and Dar [14, 15] proposed a new considered fuzzyinformation measure of order 𝛼 and type 𝛽:

𝐻𝛽

𝛼(𝐴; 𝑈)

=

121−𝛼 − 1

[

[

∑𝑛

𝑖=1 𝑢𝑖𝛽{𝜇𝛼+𝛽−1𝐴

(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛼+𝛽−1

}

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))}

𝛽

− 1]

]

.

(9)

And for this information measure they introduced the givenmean code length of order 𝛼 and type 𝛽:

𝐿𝛽

𝛼(𝑈)

=

121−𝛼 − 1

[

[

[

{

{

{

∑𝑛

𝑖=1 𝑢𝑖𝛽{𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽}𝐷((1−𝛼)/𝛼)𝑙

𝑖

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))}

𝛽

}

}

}

𝛼

− 1]]

]

.

(10)

Choudhary and Kumar [16] proved some noiseless codingtheorem on generalized R-Norm entropy. Also, Choudharyand Kumar [17] proposed some coding theorems on gen-eralized Havrda-Charvat and Tsallis entropy. Baig and Dar[14, 15] introduced few coding theorems on fuzzy entropyfunction depending upon parameters R andV and gave fuzzycoding theorem on generalized fuzzy cost measure. Tanejaand Bhatia [18] proposed a generalized mean codewordlength for the best 1:1 code and Parkash and Sharma [19, 20]proved some noiseless coding theorems corresponding tofuzzy entropies and introduced a new class of fuzzy codingtheorems. Parkash [21] introduced a new parametricmeasureof fuzzy entropy. Gupta et al. [22] proposed 1:1 codes forgeneralized quantitative-qualitative measure of inaccuracy.

International Journal of Mathematics and Mathematical Sciences 3

Jain and Tuteja [23] introduced a coding theorem connectedwith useful entropy of order 𝛼 and type 𝛽. Tuli [24] intro-duced mean codeword lengths and their correspondencewith entropy measures. Tuli and Sharma [25] proved somenew coding theorems and consequently developed some newweighted fuzzy mean codeword lengths corresponding to thewell-known measures of weighted fuzzy entropy.

In the next section, we will prove the fuzzy noiselesstheorem for 1:1 codes of binary size and hence prove that theyare less constrained.

2. Fuzzy Noiseless Coding Theorem for1:1 Codes

Theorem 1. Themean codeword length order 𝛼 and type 𝛽 for1:1 binary code satisfy:

𝐻𝛽

𝛼(𝐴; 𝑈) ≤ 𝐿 (1 :1)𝛽

𝛼(𝑈) , (11)

where

𝐻𝛽

𝛼(𝐴; 𝑈) =

121−𝛼 − 1

[

[

∑𝑛


(𝑥𝑖) + (1 − 𝜇


}

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))}

𝛽− 1]

]

,

𝐿 (1 :1)𝛽𝛼(𝑈) =

121−𝛼 − 1

[

[

[

{

{

{

∑𝑛


𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽} 2((1−𝛼)/𝛼) log((𝑖+2)/2)

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))}

𝛽

}

}

}

𝛼

− 1]]

]

.

(12)

Proof. Consider the following fuzzy information measure:𝐻𝛽

𝛼(𝐴; 𝑈)

=

121−𝛼 − 1

[

[

∑𝑛


(𝑥𝑖) + (1 − 𝜇


}

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))}

𝛽

− 1]

]

.

(13)

And hence corresponding to the above fuzzy informationmeasure we introduce the given mean codeword length for1:1 code as follows:

𝐿 (1 :1)𝛽𝛼(𝑈) =

121−𝛼 − 1

[

[

[

{

{

{

∑𝑛


𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽} 2((1−𝛼)/𝛼) log((𝑖+2)/2)

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))}

𝛽

}

}

}

𝛼

− 1]]

]

. (14)

Using, Holder’s inequality,

𝑛

∑

𝑖=1𝑥𝑖𝑦𝑖≥ (

𝑛

∑

𝑖=1𝑥𝑝

𝑖)

1/𝑝

(

𝑛

∑

𝑖=1𝑦𝑞

𝑖)

1/𝑞

. (15)

For all, 𝑥𝑖, 𝑦𝑖> 0, 𝑖 = 1, 2, . . . , 𝑛; 1/𝑝 + 1/𝑞 = 1, 𝑝 < 1( = 0),

𝑞 < 0 or 𝑞 < 1( = 0), and 𝑝 < 0.Let 𝑝 = (𝛼 − 1)/𝛼, 𝑞 = 1 − 𝛼,

𝑥𝑖=

(𝑢𝑖{𝜇𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))})𝛼𝛽/(𝛼−1) 2−log((𝑖+2)/2)

∑𝑛

𝑖=1 (𝑢𝑖 {𝜇𝐴 (𝑥𝑖) + (1 − 𝜇𝐴 (𝑥𝑖))})𝛼𝛽/(𝛼−1) ,

𝑦𝑖

=

𝑢𝛽/(1−𝛼)𝑖

(𝜇(𝛼+𝛽−1)/(1−𝛼)𝐴

(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))(𝛼+𝛽−1)/(1−𝛼)

)

∑𝑛

𝑖=1 (𝑢𝑖 {𝜇𝐴 (𝑥𝑖) + (1 − 𝜇𝐴 (𝑥𝑖))})𝛽/(1−𝛼) .

(16)

Then, (15) becomes

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛽−1𝐴

(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽−1} 2−log((𝑖+2)/2)

∑𝑛

𝑖=1 𝑢𝛽


(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽−1}

≥[

[

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽} 2((1−𝛼)/𝛼) log((𝑖+2)/2)

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽}

]

]

𝛼/(𝛼−1)

⋅[

[

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛼+𝛽−1𝐴

(𝑥𝑖) + (1 − 𝜇


}

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


]

]

1/(1−𝛼)

.

(17)


Now, we use the following condition:

𝑛

∑

𝑖=1𝑢𝛽


(𝑥𝑖) + (1−𝜇

𝐴(𝑥𝑖))𝛽−1} ⋅𝐷−𝑙𝑖

≤

𝑛

∑

𝑖=1𝑢𝛽

𝑖{𝜇𝛽

𝐴(𝑥𝑖) + (1−𝜇

𝐴(𝑥𝑖))𝛽} .

(18)

Thus, the equation is reduced as follows:

[

[

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽} 2((1−𝛼)/𝛼) log((𝑖+2)/2)

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


]

]

𝛼/(1−𝛼)

≥[

[

∑𝑛

𝑖=1 𝑢𝛽


(𝑥𝑖) + (1 − 𝜇


}

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


]

]

1/(1−𝛼)

.

(19)

Now, we raise the power on both sides by (1 − 𝛼), and we get

[

[

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽} 2((1−𝛼)/𝛼) log((𝑖+2)/2)

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


]

]

𝛼

− 1 ≥∑𝑛

𝑖=1 𝑢𝛽


(𝑥𝑖) + (1 − 𝜇


}

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


− 1.

(20)

Hence, we multiply both sides by 1/(21−𝛼 −1) > 0, and we get

121−𝛼 − 1

[

[

[

{

{

{

∑𝑛


𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽} 2((1−𝛼)/𝛼) log((𝑖+2)/2)

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))}

𝛽

}

}

}

𝛼

− 1]]

]

≥

121−𝛼 − 1

[

[

∑𝑛


(𝑥𝑖) + (1 − 𝜇


}

∑𝑛

𝑖=1 𝑢𝛽

𝑖{𝜇𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))}

𝛽− 1]

]

.

(21)

That is, 𝐿(1 :1)𝛽𝛼(𝑈) ≥ 𝐻

𝛽

𝛼(𝐴; 𝑈).

Hence, 𝐻𝛽𝛼(𝐴; 𝑈) ≤ 𝐿(1 :1)𝛽

𝛼(𝑈); this proves the

result.

Theorem 2. Themean codeword length order 𝛼 and type 𝛽 for1:1 binary code satisfy the following:

𝐿 (1 :1)𝛽𝛼(𝑈) < 𝐻

𝛽

𝛼(𝐴; 𝑈) 21−𝛼 + 1. (22)

Proof. We have

2− log((𝑖+2)/2) ≤𝜇𝛼

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛼

(∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛼+𝛽−1𝐴

(𝑥𝑖) + (1 − 𝜇


)) / (∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽))

. (23)

That is,

log(𝑖 + 22

) ≤ log(𝜇𝛼

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛼

(∑𝑛

𝑖=1 𝑢𝛽


(𝑥𝑖) + (1 − 𝜇


)) / (∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


)

−1

⋅ 2 (24)

International Journal of Mathematics and Mathematical Sciences 5

or

2log((𝑖+2)/2) < (𝜇𝛼

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛼

(∑𝑛

𝑖=1 𝑢𝛽


(𝑥𝑖) + (1 − 𝜇


)) / (∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


)

−1

⋅ 2 (25)

or

2((1−𝛼)/𝛼) log((𝑖+2)/2) < (𝜇𝛼

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛼

(∑𝑛

𝑖=1 𝑢𝛽


(𝑥𝑖) + (1 − 𝜇


)) / (∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


)

(𝛼−1)/𝛼

⋅ 2(1−𝛼)/𝛼.

(26)

Hence, we multiply both sides by (𝑢𝛽

𝑖(𝜇𝛽𝐴(𝑥𝑖) + (1 −

𝜇𝐴(𝑥𝑖))𝛽))/(∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽)).

And summing over 𝑖 = 1, 2, . . . , 𝑛, we get

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽 2((1−𝛼)/𝛼) log((𝑖+2)/2))

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽)

< (

∑𝑛

𝑖=1 𝑢𝛽


(𝑥𝑖) + (1 − 𝜇


)

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


)

1/𝛼

⋅ 2(1−𝛼)/𝛼

(27)

or

(

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽 2((1−𝛼)/𝛼) log((𝑖+2)/2))

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


)

𝛼

< (

∑𝑛

𝑖=1 𝑢𝛽


(𝑥𝑖) + (1 − 𝜇


)

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


) ⋅ 21−𝛼

(28)

or

(

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽 2((1−𝛼)/𝛼) log((𝑖+2)/2))

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


)

𝛼

− 1

< (

∑𝑛

𝑖=1 𝑢𝛽


(𝑥𝑖) + (1 − 𝜇


)

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


− 1)

⋅ 21−𝛼 + 21−𝛼 − 1.

(29)

Since 21−𝛼 − 1 > 0 for 0 < 𝛼 < 1 and after suitable operations,we get

121−𝛼 − 1

[

[

[

(

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇

𝐴(𝑥𝑖))𝛽 2((1−𝛼)/𝛼) log((𝑖+2)/2))

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


)

𝛼

− 1]]

]

<

121−𝛼 − 1

[

[

∑𝑛

𝑖=1 𝑢𝛽


(𝑥𝑖) + (1 − 𝜇


)

∑𝑛

𝑖=1 𝑢𝛽

𝑖(𝜇𝛽

𝐴(𝑥𝑖) + (1 − 𝜇


− 1]

]

⋅ 21−𝛼 + 21−𝛼 − 121−𝛼 − 1

(30)


or we can write 𝐿(1 : 1)𝛽𝛼(𝑈) < 𝐻

𝛽

𝛼(𝐴; 𝑈)21−𝛼 + 1, which

proves the result.

3. Conclusion

In modern data transmission and storage systems, the keyingredients that help in achieving the high degree of reliabilityare the error correcting codes. The noiseless channel facesthe problem of efficient coding of messages and hence wehave to maximize the number of messages to be sent througha channel in a given time. Thus, we have introduced meancodeword length of order 𝛼 and type 𝛽 for 1:1 codes andanalyzed the relationship between average codeword lengthand fuzzy informationmeasures for binary 1:1 codes. Further,noiseless coding theorems associated with fuzzy informationmeasure have been established.

4. Future Research Endeavors

Decision-making problems in general utilize informationfrom the knowledge base of experts in view of their per-ception about that system. Besides mathematical studies,algorithms for application in decision-making can also bediscussed in extension of the above research. Comparisonof other fuzzy information measures in the light of fuzzynoiseless theorem and 1:1 codes of binary size can also beproved and compared. Further, this fuzzy noiseless codingtheorem can be utilized for characterizing and applying R-Norm entropy measures.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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