ABSTRACT The strength and

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1 VOL NO Mach 0 ARPN Joal of Sciece ad Techology 0-0 All ight eeed ISSN oeig Radi of RM Biay ode O PViocha JSBhlla BSBa Feozep ollege of Egieeig ad Techology Feozep Pjab Idia Depatet of Applied Sciece Malot Ititte of Maageet ad Ifoatio Techology MIMIT Malot Pjab Idia Depatet of Applied Sciece Baba Faid ollege of Egieeig ad Techology Bathida Pjab Idia ABSTRAT The tegth ad the coeig adi of a biay elf-copleetay code ae tdied de diffeet coditio Soe elt fo coeig adi ae fod with diffeet age of ale of weight of code We hae geealized the elt coceig elatio betwee tegth ad coeig adi of code The we hae poed the elt fo biay code of legth haig tegth whichh epeet the elatio aog weight of codewod ad be of codewod i code Fthe we hae aalyzed the paticla elt by ig thi elatio fo code haig tegth oe tha ie fo tegth = ad obtaied the age of ale of coeig adi Latly we hae aalyzed the ale of coeig adi ad fod that i cae of ee ffix the ale ae clea-ct fixed; bt i cae of odd ffix oly ale of ae fixed bt aftewad the ale becoe oe ad oe catteed ad poble of fixig iqe ale becoe oe ad oe difficlt Keywod: Reed-Mlle code biay elf-copleetay code coet weight tegth coeig adi elatio geealizatio INTRODUTION The coeig adi of a code GF i that it i the leat itege = ch that each ecto i GF i withi Haig ditace of oe codewod i Delate 97[] poed that i le tha o eqal to the exteal ditace of whee exteal ditace of code i the be of o-zeoo te i the MacWillia tafo of ditace ditibtio ditibtio of code Alo we defiee the coeig adi of code to be t = axidit fff So t i the te exteal ditace / of code ie t i the axi of the allet weight i ay coet of Alo a M d = e + code i pefect iff / = e e beig the eo-ecto MacWillia ad Sloae977[] Theefoe if a code i a e-pefect code the it i clea that = e I paticla if a code i oe-eo-coectig BH code of legth = - the t = ad hece code will hae coeig adi Goetei Peteo ad Ziele 90[] poed that the two-eo-coectig BH code of legth - ha coeig adi Bege ad Vade Hot fo 0od 97[] A ad Matto 97fo odd[] ad Helleeth97 fo od [] poed that the thee- eo-coectig BH code of legth - ha coeig adi HelleethT; Kloe T ad Mykkeltit J 97[7] tdied ppe bod o the coeig adi of biay code ad clae of code that attai thee bod RELATION BETWEEN STRENGTH AND OVERING RADIUS OF A ODE A biay code ha tegth if each -bet of the co-odiate of the code cotai all biay -tple a cotat be of tie Johae J Mykkeleit 90[] defied that code ha tegth iff eey bet of col of i LI ie if ha o o-zeoo codewod of weight Alo if code ha tegth the it alo hae tegth / fo all itege / ch that 0 / Lea : Let be a biay code of legth Let GF The: i If ha tegth the w ii w wheee deote thee be of codewod i code Theoe : If code i a biay code of legth ad tegth = the Now befoe poceedig to Theoe we hae a defiitio a follow: = GF If ha tegth the F ; = + + fo GF

2 VOL NO Mach 0 ISSN -77 ARPN Joal of Sciece ad Techology 0-0 All ight eeed Theoe : If i a biay epetitio code o a hoteed Hadad code of legth which i obtaied fo a oalized Hadad atix by deletig the fit col ad chagig + to 0 ad - to the = Now epetitio code i elf-copleetay ad the hoteed Hadaad code ha tegth ad i Theoe oe o the othe coditio hold ad we ee that i elatio to bod thee i o ipoeet i Theoe a copaed to the bod i Theoe Theefoe by eig both thee coditio iltaeoly we wold like to aalye whethe thee i oe ipoeet i the bod o ot I thi light we adace the dicio i the fo of Theoe Theoe : Let be a biay elf-copleetay code of legth ad tegth = the ale of i: ad appoxiate ale of i : whee 0 i If w = w < the: < / If w < w = the: < / Poof: Let GF be a ecto Let ditace of to ay codewod of code i at leat ie let w fo all + i If w = w = the: = / ; ii If w > w < the the age of the ale of i: ad appoxiate ale of i: whee 0 iii If w < w > > the the age of If = + c + the = + c + becae i copleetay theefoe if c the c =c + ϵ Alo w + w = Moeoe if w fo all + the w ; ad ice-ea 7 i Let w = w = Let take 0 Now w + w =w +-w ig = +- =

3 VOL NO Mach 0 ISSN -77 ARPN Joal of Sciece ad Techology 0-0 All ight eeed ig = [ ] = [ ] Theefoe w + w = [ ] 9 = Becae the tegth of code i theefoe applyig Lea patii we hae: w Thi iplie: 0 ig9 ad ig the fact that tegth of code i which iplie that cotibtio of each poitio i codewod of code will be / whee deote the be of codewod i code Now i: 0 Theefoe = ig Becae beig coeig adi of code i a itege theefoe iplie that: = / ii Let w > w < Now w + w < w + becae w < < + becae w + w = iplie that w < = ig Theefoe w + w = Becae the tegth of code i theefoe applyig Lea patii we hae: w Thi iplie: ig ad ig the fact that tegth of code i which iplie that cotibtio of each poitio i codewod of code will be / whee deote the be of codewod i code 0 Thi gie the age of the ale of Alo becae i ey all o will be o all ad it ca be eglected So we get fo a: 0 Thi gie the appoxiate ale of w < the the age of the ale of i: 7 Hece if w >

4 VOL NO Mach 0 ISSN -77 ARPN Joal of Sciece ad Techology 0-0 All ight eeed ad appoxiate ale of i : iii Poceedig exactly a i pat ii aboe we ca etablih that if w < w > the the age of the ale of i: ad a appoxiate ale of i : i Now we dic the cae whe w=w < Now w + w = + w becae w = < + becae w < < + - becae <- = ig Theefoe w + w < [ ] Becae the tegth of code i theefoe applyig Lea patii we hae: w Thi iplie: 9 ig ad ig the fact that tegth of code i which iplie that cotibtio of each poitio i codewod of code will be / whee deote the be of codewod i code 0 Now i: 0 Theefoe itege < ig 0 Becae beig coeig adi of code i a Theefoe iplie that: = / Poceedig exactly a i pat i aboe we ca aalye the cae whe w < w = ad aie at the coclio that < / GENERALISATIONS OF THE RESULTS ONERNING RELATION BETWEEN STRENGTH AND OVERING RADIUS OF A ODE By Lea if be a biay code of legth ad ha tegth = ad GF the: w Ad if f ha tegth = the w We dic the geealizatio of aboe elt i the fo of followig Theoe Theoe : Let be a biay code of legth Let GF If ha tegth w the Poof: lealy + = ie ad + will cotai ae be of codewodecto Alo tegth of code ad tegth of coet + i the ae Theefoe +

5 VOL NO Mach 0 ISSN -77 ARPN Joal of Sciece ad Techology 0-0 All ight eeed will hae tegth iff ha tegth Thi ea that each -bet of co-odiate of code ad each -bet of co-odiate of coet + will cotai all biay -tple a cotat be of tie So it will be fficiet to poe the Theoe fo = 0 If code ha tegth = the each -bet of co-odiate of code will cotai biay tple a cotat be of tie So each poitio will hae a eqal be of zeo ad oe Hece each poitio will cotibte / to the If code ha tegth = the each -bet of co-odiate of code will cotai biay -tple a cotat be of tie So will occ / o / tie i two fixed poitio Agig i thi way we will ee that if code ha tegth the tie will occ / tie i fixed poitio Becae legth of code i o each codewod i code will hae poitio ad be of way of coideig fixed poitio aog poitio will be Theefoe weight of code will be gie by / O the othe had weight of code will alo be gie by w A a elt we hall obtai: w = / ie w = The elt of Theoe lead to the elt of Lea a how below If code ha tegth = the elt will becoe a: w w which i the fit elt of Lea ie elt If code ha tegth = the elt will becoe a: w w w w w w ig elt of Lea pati w which i the d elt of Lea ie the elt Theefoe we ee that elt of Theoe ie the elt lead to elt i ad ii of Lea Thi fact fthe tegthe the atheticity of Theoe Now we ca aalye the paticla elt by ig elt of Theoe fo the code haig tegth oe tha Let code ha tegth ie let = Theefoe we obtai fo elt of Theoe a : w w!! w!!! 7

6 VOL NO Mach 0 ISSN -77 ARPN Joal of Sciece ad Techology 0-0 All ight eeed w w w w w w ig ad Theefoe w Let code ha tegth ie let = Theefoe we obtai fo elt of Theoe a : w!! w w w w w w w ig ad Theefoe: w Let code ha tegth ie let = Theefoe we obtai fo elt of Theoe a : w!! w w w 0 0 w w w w w So 0 0 w ig ad Theefoe: w Ad o o I thi way we ca dic the elatio betwee weight of code ad it tegth ad ca hae folatio fo weight of code i te of legth of

7 VOL NO Mach 0 ISSN -77 ARPN Joal of Sciece ad Techology 0-0 All ight eeed 9 code ad be of codewod i the code fo aio ale of tegth of code Now aog the aio cae i Theoe we ote that whe w = w = the = / ;whe w = w < o whe w < w = the < / obiig thee two cae we hae: / I the poce of poof of thee cae we coe aco folatio 0 ad 9 ad cobiig thee we obtai: w w o The tetatie geealizatio of thi ay be like thi: Theoe : If code i of tegth ad i elf- copleetay the: w If tegth of code i the will becoe a: w That i: w 9 If tegth of code i the will becoe a: w w 0 If tegth of code i the will becoe a: w ig 0 Theefoe whe code i of tegth = ad i elf-copleetay the: If tegth of code i the will becoe a:

8 VOL NO Mach 0 ISSN -77 ARPN Joal of Sciece ad Techology 0-0 All ight eeed 0 w ig 0 Now ole the eqatio: 0 9 Now caot be eqal to 9 a i alway poitie Theefoe 9 9 Let : iplie Theefoe 0 : iplie Now Theefoe If i ot a pefect qae the: Theefoe iplie: If i a pefect qae ie if = ay o that iplie that the: ad theefoe : 7 If tegth of code i the will becoe a: w

9 VOL NO Mach 0 ISSN -77 ARPN Joal of Sciece ad Techology 0-0 All ight eeed ig 7 So 0 Now we ole the eqatio which i qadatic i ρ : 0 Now caot be eqal to a i alway poitie Theefoe Let 9 : iplie Theefoe 0 0 : 9 iplie Now Theefoe If i ot a pefect qae the: Theefoe iplie: If i a pefect qae ie if = ay o that iplie that the: ad theefoe: So we hae the followig Theoe: Theoe : Let i a biay elf-copleetay code of legth ad of tegth o i If i ot a pefect qae the: ii If i a pefect qae ay = ay the: PROBLEM OF OVERING RADIUS WITH ODD SUFFIX Fo the coet weight ditibtio of RM i kow We kow that 0 = 0 = 0 = = = = Now we hae a Theoe decibig geeal folatio fo a follow:

10 Theoe 7: Fo 0 VOL NO Mach 0 ISSN -77 ARPN Joal of Sciece ad Techology 0-0 All ight eeed / Now i Theoe we ee that whe w= w = the = / ; whe w = w < the < / / ; whe w < w = the < obiig thee we ee fo Theoe that if i a biay elf-copleetay code of legth ad tegth ad whe w= w = o whe w = w < o whe w < w = the / By Theoe we ee that fo 0 / obiig thee two Theoe we obtai: / / & becae i Theoe legth of code i theefoe i becoe a = ie / / & ie / / & ie / & ie & ie - - & becae i a itege ie - - Pttig ale of =0 we obtai fo : 0 = - - = 0 = - 0 = - = = = - = = = - = = 7 = - =0ad o o Now it i obio that: > fo By Theoe we hae: / ad by Theoe we hae: / Theefoe + ie + ie + ie + ie + / fo Theefoe fo & we obee that becae ad ae ot eqal fo 0 o we caot cobie thee two elt i to obtai a igle elt fo + We obtai fo : Uppe bod of + i ad Lowe bod of i fo 7 Pttig = i 7 we get: Uppe bod of = =; ad Lowe bod of = which i the kow ale of Pttig = i 7 we get: Uppe bod of 7 = = = =; ad Lowe bod of = = = = = 7 Theefoe 7 Hece it i difficlt to fix the ale of 7 a o 7 o Pttig = i 7 we get:

11 VOL NO Mach 0 ISSN -77 ARPN Joal of Sciece ad Techology 0-0 All ight eeed Uppe bod of 9 = = = 7 = ; ad Lowe bod of = 9 = 0 Theefoe 0 9 Hece it i difficlt to fix the ale of 9 a 0 o o o o Pttig = i 7 we get: Uppe bod of = 0 = 0 =0 = 9 00; ad Lowe bod of = = 0 Theefoe Hece it i difficlt to fix the ale of a 99 o 99 o 99 o 99 o 99 o 997 o 99 o 999 o 000 o 00 Ad o o Fo thi dicio we coclde that i cae of 0 the ale ae clea-ct fixed Bt i cae of 7 9 we obee that oly ale of ae fixedbeig 0 ad epectiely ad afte that i cae of 7 9 the poble of fixig iqe ale of thee becoe oe ad oe difficlt =0 0 = 0 = = = 7 = 0 9 = = 9 So i cae of coeig adi with ee ffix thee i o poble of fixig the ale Bt i cae of coeig adi with odd ffix the ale of ae fixed bt fo owad the poble goe o becoig oe ad oe difficlt we get oe ad oe catteed ale ONLUSION Thee i a cloe elatio betwee tegth ad coeig adi ad betwee coeig adi ad weight of a biay elf-copleetay code Fo diffeet age of weight of code we hae obtaied diffeet age of ale of coeig adi The elt coceig elatio betwee tegth ad coeig adi of code ca be geealized The age of ale of coeig adi chage with the diffeet ale of tegth of code Alo if tegth of code i o the ale of coeig adi deped po whethe legth of code i a pefect qae o ot The ale of coeig adi with ee ffix ae fixed; bt with odd ffix i cae of ad the ale ae fixed bt aftewad the age of ale becoe oe ad oe catteed REFERENES [] P Delate 97: Fo Fdaetal Paaete of a ode ad thei obiatoial Sigificace Ifo otol pp [] MacWillia FJ ad Sloae NJA977: The Theoy of Eo-oectig ode Ateda: Noth Hollad 977 [] Goetei D Peteo W ad Ziele N90: Two-Eo oectig Boe-hadhi ode ae Qai-Pefect Ifo otol pp [] A J EF ad Matto J HF 97: Soe Thee-Eo oectig BH ode hae oeig Radi IEEE Ta Ifo Theoy Vol IT- pp-9 97 [] Bege T ad Vade Hot JA 97: oplete Decodig of Tiple-Eo oectig Biay BH ode IEEE Ta Ifo Theoy Vol IT- pp-7 97 [] HelleethT 97: All Biay Thee-Eo- oectig BH ode of Legth - hae oeig Radi IEEE Ta Ifo Theoy Vol IT-pp 7-97 [7] HelleethT; Kloe T; Mykkeltit J97: O the oeig Radi of Biay ode IEEE Taactio o Ifoatio Theoy Vol IT-No pp7- Septebe97 [] Mykkeltit J90: The oeig Radi of the Reed-Mlle ode i IEEE Taactio o Ifoatio Theoy Vol IT- No pp9- May90

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