CONTRIBUTION OF KRAFT S INEQUALITY TO CODING THEORY

Transcription

1 Pacfc-Asa Joura of Mathematcs, Voume 5, No, Jauary-Jue 20 CONTRIBUTION OF KRAFT S INEQUALITY TO COING THEORY OM PARKASH & PRIYANKA ABSTRACT: Kraft s equaty whch s ecessary ad suffcet codto for the exstece of a uquey decherabe code ays a mortat roe the terature of codg theory I the reset commucato, we have defed the Kraft umbers, ad usg ths stadard equaty, deveoed some ew resuts whch are cosey reated wth the Kraft s equaty INTROUCTION It s we kow fact that the tw dsces of formato theory ad codg theory were the outcomes of the major dscovery by Shao [0] after the ubcato of hs frst aer etted A Mathematca Theory of Commucato The basc goa s effcet ad reabe commucato a ucooeratve (ad ossby hoste) evromet To be effcet, the trasfer of formato must ot requre a rohbtve amout of tme ad effort To be reabe, the receved data stream must resembe the trasmtted stream to wth arrow toeraces These two desres w aways be at odds, ad our fudameta robem s to recoce them as best we ca At a eary stage, the mathematca study of such questos broke to two broad areas of attracto, vz formato theory ad codg theory The frst dsce s the study of achevabe bouds for commucato ad s argey robabstc ad aaytc ature whereas the other dsce attemts to reaze the romse of these bouds by modes whch are costructed through may agebrac meas I codg theory, Kraft s [5] equaty, amed after Leo Kraft, gves a ecessary ad suffcet codto for the exstece of a uquey decodabe code for a gve set of codeword egths Its acatos to refx codes ad trees ofte fd use comuter scece ad formato theory More secfcay, Kraft s equaty mts the egths of codewords a refx code: f oe takes a exoeta fucto of each egth, the resutg vaues must ook ke a robabty mass fucto Kraft s [5] equaty ca be thought of terms of a costraed budget to be set o codewords, wth shorter codewords beg more exesve The foowg fudameta reates aways hod for Kraft s equaty: () If Kraft s equaty hods wth strct equaty, the code has some redudacy Keywords: Uquey echerabe code, Istataeous code, Kraft s equaty, Etroy, Mea codeword egth, Covex mag

2 46 Om Parkash & Pryaka () If Kraft s equaty hods wth strct equaty, the code questo s a comete code () If Kraft s equaty does ot hod, the code s ot uquey decodabe Kraft s equaty dscusses oy refx codes, ad s sometmes aso caed the Kraft McMa theorem after the deedet dscovery of the resut by McMa [8] who roved the resut for the geera case of uquey decodabe codes Nagaraja [9] oted out that the Kraft McMa equaty s a basc resut formato theory whch gves a ecessary ad suffcet codto for a code to be uquey decodabe ad aso has a quatum aaogue Some other cotrbutos towards Kraft s equaty have bee made by Ludwg [7] ad aa [2] Kraft s [5] equaty exteded tremedous acatos the fed of codg theory ad was used by Shao [0] ad Cambe [] for deveog the ower bouds for the arthmetc mea ad exoetated mea resectvey Some other oeer who worked for the costructo of codeword egths ad cosequety roved ew oseess codg theorems are Logo [6], Guasu ad Pcard [3], Gurda ad Pessoa [4] etc The foowg theorem s we kow the terature of codg theory: Theorem : (Kraft s Theorem) () If C s a -ary stataeous code wth codeword egths,,, the these egths must satsfy Kraft s equaty = () (2) If the umbers,, ad satsfy Kraft s equaty (), the there s a stataeous -ary code wth codeword egths,, Note: Kraft s [5] equaty s aso ecessary ad suffcet codto for the exstece of a uquey decherabe code Of course, Kraft s equaty s suffcet sce ay stataeous code s aso uquey decherabe The ecessty of Kraft s equaty was roved by McMa [8] as stated the foowg theorem: Theorem 2: (McMa s Theorem) If C = (c,, c ) s a uquey decherabe -ary code, the ts codeword egths must satsfy Kraft s equaty () We, ow defe for a -ary code C havg the code word egths,,, the Kraft umbers gve by the foowg mathematca exresso: K (,,,) = (2) 2 =

3 Cotrbuto of Kraft s Iequaty to Codg Theory 47 The objectve of the reset aer s to deveo some ew formato theoretc equates frequety used the terature of formato theory These equates are cosey reated wth the we kow Kraft s [5] equaty ad have bee deveoed by usg the Kraft umbers defed equato (2) I the ext secto, we deveo such equates 2 GENERATING NEW INFORMATION THEORETIC RESULTS THROUGH KRAFT S INEQUALITY Before rovg the ma resut, we frsty rove the foowg Lemma: Lemma 2: Let,, be rea umbers wth >, 0 ad The we have the doube equaty og og = = = = = The equaty hods ff = og = = (2) Proof: The exoeta ma f : R (0, ), f (x) = x, s strcty covex o R For a covex mag f whch s dfferetabe o ts doma, we have the doube equaty for a x, y the doma of f f ()()()()()() y x y f x f y f x x y, (22) As f (x) = x, the by (22), we get y x y x () x() y, x y x y R, (23) Now f we choose to the equaty (23), x =, y = og, we get for a (, 2,, ) ( og) ( og), (24) Summg (24) over from to, we have = = = = ( og) ( og) og og (25) = = = = = The case of equaty foows by strct covexty of the mag f(x) = x, ( >,x R)

4 48 Om Parkash & Pryaka Theorem 22: If L = s the average code-word egth of a uquey = decherabe code for the radom varabe X, the L H (X) wth equaty ff = og where H () X = og s the Shao s etroy = Proof: Sce code s uquey decherabe, we have from Kraft s equaty The from (25), we have = og 0 = = that s, L H (X) or og = = The case of equaty s obvous by the above emma Theorem 23: Let C = (c,, c ) be a -ary code havg the codeword egths,, The we have the estmato for the Kraft umbers: = 2 (() ) og K(, 2,,) (26) = The equaty hods ff = og Proof: By emma 2, we have = + og = = = = + og = = = = og

5 Cotrbuto of Kraft s Iequaty to Codg Theory 49 ad = + og = = = = + og = = = 2 (() ) = The case of equaty s obvous by the same emma Coroary 24: Let C = (c,, c ) be a -ary code havg the codeword egths,, If the C s ot uquey decherabe og (27) = = Coroary 25: If the rea umbers,, ( =, 2,, ) wth >, 0 ad = satsfy the equaty = = = og ` the there s a stataeous code -ary code wth codeword egths,, Proof: Note that the equaty (28) s ceary equvaet wth = = but by the equaty (2), we have ad the = og 0 0 og = = = Ayg Kraft s theorem, we get the desred resut (28)

6 50 Om Parkash & Pryaka Lemma 26: Let,, be rea umbers wth >, 0 ad The we have the doube equaty ( ) ( ) = = = = = (29) The equaty hods ff = og Proof: The mag g (x) = x, s strcty covex o (0, ) so by the equaty (22), we have the equaty b ()() a b a b a a b (20) for a a, b [0, ) Let us choose (20), =, a =,() b = to get for a {, 2,, } whch s equvaet wth Summg (2) over from to, we get (2) = = = = = = = = = = = whch s the desred resut (29) The case of equaty hods from the strct covexty of g ad takg to accout that = ff og = e, = og for a {, 2,, }

7 Cotrbuto of Kraft s Iequaty to Codg Theory 5 Theorem 27: Let C = (c,, c ) be a -ary code havg the codeword egths,, The, we have the foowg estmato for the Kraft umbers: ( ( )) ( 2 ),,, K + = = (22) The equaty hods ff = og Proof: The roof foows from the above emma Proosto 28: Let C = (c,, c ) be a -ary code havg the codeword egths,, If, = = (23) the C s ot uquey decherabe Proof: Let us suose that C s uquey decherabe, the by McMa s theorem, we have = Usg emma (26) ad (24), we have (24) = = ( ) 0 whch = = gves ad ths cotradcts (23) Hece the resut Theorem 29: If the rea umbers,, ( =, 2,, ) wth >, 0 ad = satsfy the equaty =

8 52 Om Parkash & Pryaka = = (25) the there s a stataeous code -ary code wth codeword egths,, Proof: The equaty (25) s ceary equvaet to But by equaty (29), we have = = 0 = = = 0 whch gves = Ayg Kraft s theorem, we arrve at the desred resut ACKNOWLEGEMENTS The authors are thakfu to Uversty Grats Commsso ad Couc of Scetfc ad Idustra Research, New eh, for rovdg the faca assstace for the rearato of the mauscrt REFERENCES [] Cambe L L, A Codg Theorem ad Rey s Etroy, Iformato ad Cotro, 8, (965), [2] aa M, O Uque ecodabty, IEEE Tras Iform Theory, 54(), (2008), [3] Guasu S, ad Pcard C F, Bore Ifercutre de a Loguerur ute de Certa Codes, C R Acad Sc, Pars, 273A, (97), [4] Gurda, ad Pessoa F, O Usefu Iformato of Order α, J Comb Iformato ad Syst Sc, 2, (977), [5] Kraft L G, A evce for Quatzg Groug ad Codg Amtude Moduated Puses, MS Thess, Eectrca Egeerg eartmet, MIT, (949) [6] Logo G, A Noseess Codg Theorem for Sources Havg Uttes, SIAM J A Math, 30(4), (976),

9 Cotrbuto of Kraft s Iequaty to Codg Theory 53 [7] Ludwg, S (2007): O maxma refx codes, Bu Eur Assoc Theor Comut Sc, 9, (2007), [8] McMa B, Two Iequates Imed by Uque echerabty, IRE Tras If Theory, 2(4), (956), 5 6 [9] Nagaraja N, A yamca Systems Proof of Kraft McMa Iequaty ad Its Coverse for Prefx-Free Codes, Chaos, 9, (2009), 0336() 0336(5) [0] Shao C E, A Mathematca Theory of Commucato, Be System Tech J, 27, (948), Om Parkash eartmet of Mathematcs, Guru Naak ev Uversty Amrtsar Ida, E-ma: omarkash777@yahooco Pryaka eartmet of Mathematcs Guru Naak ev Uversty, Amrtsar Ida, E-ma: ryaka_kakkar85@yahoocom