Coding Theorems on New Fuzzy Information Theory of Order α and Type β

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1 Progress Noear yamcs ad Chaos Vo 6, No, 28, -9 ISSN: oe Pubshed o 8 February 28 wwwresearchmathscorg OI: Progress Codg Theorems o New Fuzzy Iormato Theory o Order ad Type Saeea Peerzada, Sama Mazoor So 2, MK Bag 3 ad shq ussa Bhat 4 Post-Graduate epartmet o Statstcs, Uversty o Kashmr, Sragar96, Ida E-ma: sapezad@gmacom; 2 E-ma:samamstsc@gmacom 3 E-ma: bagmak@gmacom; 4 E-ma: ashqhb4@gmacom; Correspodg uthor Receved 28 Jauary 28; accepted 2 February 28 bstract I ths paper, we propose a ew two parametrc uzzy average code-word egth or a uzzy set ad ts reatoshp wth uzzy ormato measure s dscussed so, the bouds o proposed average code-word egth, terms o uzzy ormato measure, are obtaed Keywords: Codg Theorem, Code-Word egth, oder s Iequaty, ad Krat s Iequaty MS Mathematcs Subject Casscato 2: 947, 9424 Itroducto ue to ack o sharp dstcto whether a partcuar tem beogs to a set or ot, a cocept o mperect ormato arses e, uzzess The cocept o uzzy sets troduced by Zadeh [] uses mprecse kowedge to dee a evet uzzy subset o uverse X {, 2,, } s deed as {, : X, [,] &,2, }, s a membershp ucto whch gves the degree o beoggess o the where eemet to the set I case, or or a the s caed a crsp set We take X {, 2,, } a dscrete radom varabe wth respectve probabtes p,, p2, p, p ad p Shao [9] troduced the oowg measure o ormato ad amed t etropy P p og p

2 Saeea Peerzada, Sama Mazoor So, MK Bag ad shq ussa Bhat et us cosder code words c, c2,, c o egths, 2,, wth probabtes p, p2,, p satsyg the Krat [5] Iequaty, where s the sze o code aphabet, the the ower boud o mea code word egth e, betwee P ad P as proved by Shao [9] hs oseess codg theorem The cocept o uzzess, because o ts property to cosder accurate ad ambguous ormato has vast appcabty that covers eds o egeerg, computer scece, medce, uzzy arcrat cotro ad so o eret ormato measures aog wth the basc oseess codg theorems have bee gve by severa authors who cude Kapur [3,4], Rey [7], shq eta [,2], Bag et a [6] etc The ower bouds or the average code-word egth o a uquey decpherabe code have bee obtaed terms o Shao s [9] measure o etropy Kapur [4] propouded the reato betwee the probabstc etropy ad codg The probabstc measures o etropy do ot work a stuatos; such cases the dea o uzzess ca be cosdered 2 Geerazed uzzy average code word egth Cosder the measure proposed by Saeea [8] eta as p es { } ;, & og > Correspodg to the above measure, we propose the oowg average code-word egth og { } ;, & > 2 I equato 2, s the sze o code aphabet Now, we obta the bouds o 2 terms o uder the oowg codto { } 3 Or we ca wrte 3 as {, } 4 where, 5 whch s the geerazed uzzy Krat s [5] equaty 2

3 Codg Theorems o New Fuzzy Iormato Measure o Order ad Type 3 Theorem 2 For a tegers >, the code-word egths,, 2, satsyg the codto 4, the the code-word egth 2 satses the equaty &, ; > 6 The equaty hods 6, og 7 where, Proo: By oder s equaty, we have & 2,,, ;, ; q p y y y q q p p 8, q p or, p q The equaty hods ad oy there ests a postve costat c such that q p cy Substtutg,,, y, & q p Usg these vaues 8 ad ater smpcato, we get,,, 9

4 Saeea Peerzada, Sama Mazoor So, MK Bag ad shq ussa Bhat 4 Usg equaty 4, we get,,,, ppyg ogarthms wth base to the both sdes o, we get, og, og s, mutpyg both sdes o by >, we get, og, og 2 Usg 5 2, we get og og We ca wrte the above as ece, the resut s estabshed We have rom equato 7,, og

5 Codg Theorems o New Fuzzy Iormato Measure o Order ad Type 5 Or, 3 Rasg both sdes o 3 to the power, we get Or, Or, 4 Mutpyg both sdes o equato 4 by,, ad the summg over, 2,,, we get,, 5 ppyg ogarthms to the base o both sdes o 5, ad the mutpyg both sdes by, we get, og, og We ca wrte the above equato as ece, the resut s estabshed Theorem 22 For every code wth egths,, 2, satsyg the codto 4, ca be made to satsy the equaty

6 Saeea Peerzada, Sama Mazoor So, MK Bag ad shq ussa Bhat 6 > &, ; 6 Proo: From thetheorem 2, we have 7 7 hods ad oy &, ;, >, og We chose the code-word egths 2,,, such a way that they satsy the equaty, og, og 8 Cosderg the terva o egth uty as, og,, og δ I every δ, there es oe postve teger such that, og, og 9 We w rst show that the sequece,, 2, satses the geerazed uzzy Krat [5] equaty 4 Now, takg S o 9, we have, og

7 Codg Theorems o New Fuzzy Iormato Measure o Order ad Type 7, 2 Mutpyg both sdes o 2 by, ad the summg over, 2,, o both sdes, we get, whch s the geerazed uzzy Krat [5] equaty Now, takg RS o 9, we have, og, 2 Sce, thus > ad > Now, rasg both sdes o 2 to the power, we have,, 22 Mutpyg both sdes o equato 22 by,, ad the summg over,,2, o both sdes, we get,, 23 ppyg ogarthm to the base o both sdes o 23, we get

8 og Saeea Peerzada, Sama Mazoor So, MK Bag ad shq ussa Bhat {, } og {, } Sce & thus > & > equato 24 by, we get og, og or ;, & > ece, the resut s estabshed 24 Mutpyg both sdes o {, } 3 Cocuso I ths artce, we propose a ew geerazed uzzy code word egth ad deveop the codg theorems correspodg to ths code word egth ad aso show that ;, & > REFERENCES Bshq ad MKBag, Bouds o two parametrc ew geerazed uzzy etropy, Mathematca Theory ad Modeg, Bshq ad MKBag, Some codg theorems o ew geerazed uzzy etropy o order ad type, pped Mathematcs ad Iormato Sceces etters, JNKapur, geerazato o Campbes oseess codg theorem, Joura o Bhar Mathematca Socety, JNKapur, Measures o Fuzzy Iormato, Mathematca Scece Trust Socety, New eh JKrat, devce or quatzg groupg ad codg amptude moduates puses, MSThess, epartmet o Eectrca Egeerg, MIT, Cambrdge MKBag, MBhat ad MJar, Some ew geerazatos o uzzy average code word egth ad ther bods, merca Joura o pped Mathematcs ad Statstcs, Rey, O measure o etropy ad ormato, Proceedg Fourth Berkey Symposum o Mathematca Statstcs ad Probabty, Uversty o Caora, Press,

9 Codg Theorems o New Fuzzy Iormato Measure o Order ad Type 8 SPeerzada, SMazoor, MKBag ad Bshq, ew geerazed uzzy ormato measure ad ts propertes, Iteratoa Joura o dvace Research Scece ad Egeerg, CEShao, mathematca theory o commucato, Be System Techca Joura, Zadeh, Fuzzy sets, Iormato ad Cotro,