CHAPTER-4 A BROAD CLASS OF ADDITIVE ERROR CODING, CHANNELS AND LOWER BOUND ON THE PROBABILITY OF ERROR FOR BLOCK CODES USING SK- METRIC

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1 CHATER-4 A ROAD CLASS OF ADDITIVE ERROR CODING CHANNELS AND LOWER OUND ON THE ROAILITY OF ERROR FOR LOCK CODES USING SK- METRIC Th ctts f ths Chat a basd m fllwg ublshd a: Gau A Shama D A ad Class f Addtv E Cdg Chals ad Lw ud th bablt f E f lc Cds usg SK- Mtc Itatal Jual f Ald Mathmatcs ad Statstcs vl 5 ssu

2 CHATER-4 A ROAD CLASS OF ADDITIVE ERROR CODING CHANNELS AND LOWER OUND ON THE ROAILITY OF ERROR FOR LOCK CODES USING SK- METRIC 4 INTRODUCTION I vus chats w dfd a w wa f lg f that actuall ccus dug th tasmss ad btad buds lgth ad at chcs f wl dfd atts I actc ths calls f vstg th chals whch match wth ths s mv gall wth SK-att f GF W dal wth ths blm ths chat Hammg mtc basd studs t ma b calld that Hammg mtc was tall sd f ba cds ad thb f ba chals ad that ts us was staghtfwadl tdd t -ba cass wthut lg th ssbl mtcs f -a cass Ath mtc th L-mtc was f cus als tducd a ath adhc ma ut ths basc tl th mtc mld f masug ssbl s ctud wthut d stud As mtd al ths ga was flld b Shama ad Kaush b lg all ssbl mtcs f -a cass > Th tducd a class f dstacs that cluds Hammg ad L dstacs as atcula cass Wth chcs avalabl t s w ssbl t fmulat s tms f a SK-dstac that bst ft th chal ad thus vsts th ffcc ssus ad mv th buds stg ut l ths (atal t ths ud Hammg mtc) s that actuall ha dgt-ws ath tha all that that wuld ha f Hammg dstac was csdd I 979 Kaush [6] btad Hammg ltg ad Vashamv-Glbt t f buds usg SK-dstac cta Chat-3 w hav dvd a cmbatal u bud umb f at chc dgts f la cds that cct all what w call atal adm f a ( ) cd wth mmum SKdstac at last d Ou sult galzd llustus Vashamv-Glbt bud whch fllws fm t as a atcula cas I ths Chat usg SK-mtc aach w galz th da f addtv s b tducg a bad class f Class-addtv s I th sttg f SK-dstacs ad 55

3 bablts w galz th cct f ba-smmtc-chal t what s -a-sk- att Smmtc Chal W stud th bablstc ascts f ctllg cds ad t sm sults buds bablt f f blc cds dvld th -a SK- att Chal 4 A GENERALIZATION OF INARY SYMMETRIC CHANNEL Sc SK-basd studs csd atts f csdg t a SK att { } m hav t b latd t th att whch w df as fllws: Z f dstac tc a Z chal f Z m th bablts ( j / ) DEFINITION: SK-DISCRETE MEMORYLESS CHANNEL Gv a SK-att { } m f chal has th tast bablts gv b Z m th SK-dsct mmlss ( j ) ( ) ( j( cvd) ( st) ) l l l f j Z j l m md l wh ) s bablt f sub-st l m ( l l th att { } m f Z W shall dt ths -a smmtc chal f SK-att b SC ad t wll hav chal mat gv b C ( ) ( l j / ) ( j ) / : j l l NOTE: It ma b td that th dft adtd abv th bablt s asscatd wth th class t whch (gv b j md ) all blgs Als t ma b ucl s that f ba cas ad Hammg dstac th chal ducs t a-smmtc- Chal I ths ss th dft abv galzs th SC f a cas ddg 56

4 57 u th udlg SK-att That ths s als smmtc atu ma b asl vfd ad s llustatd b a aml gv blw EXAMLE 4: Lt { } Z ad th SK-atts f 7 Z b gv b { } wh { } { } 56 ad { } 34 s that 4 ad th wth havg sctvl th bablts s that th chal mat s gv b FIGURE 4: Chal Mat j

5 Ths chal tms f tat adtd s a 7 SC wh 7 ad {()(56)(34)} 43 CODES WITH SK-ARTITION CLASS ADDITIVE ERRORS I ths sct w df s asg fm addt f scfc classs f a SKatt Ths aach galzs sval das cludg what w call adl Galzd SK-dstac fct Cds studd chat-5 - ADDITIVE ERROR CODE Wth th tduct f SK-att t s ssbl t csd vaus dfft ts f s Th s bg addtv atu as ths a f aml th ma b s that as b addt f dgts ccug a atcula attg st If th s csdd a all sts lmtd t addt f lmts f w shall call such a cd as - Addtv-E Cd DEFINITION: - ADDITIVE ERROR a v Gv a vct u ( a a ) Z ad a SK-att ( ) m th vct u wll b calld t hav -addtv s f th sts vct u udg addt f dgts th subst EXAMLE 4: Sus w hav { } att f Z3 gv b { } 3 Z wth 3 Lt us csd th SK- 3 wh { } { } { 349} ad { 5678} 3 Lt ˆ ( 547) u th û has addtv sts 3 d ad 4 th ad û bcms ( ) ( 5 4 7) 58

6 wh all addt a md 3 I gal ths da f addtv-s ma b tdd futh t addt f dgts f m tha attg st S th ca b: -Addtv s m scfcall -addtv sgl-s -addtv dubl-s m gall ddg th umb f sts whch s a lmtd scfd -addtv s f th addtv s a dgts fm u f ad tc NOTE: Wh th s a t lmtd t scfc attg st sts ths as ma b clal vsualzd wll csd t what wll atuall b th cas ladg t Hammgmtc ROADLY GENERALIZED SK-DISTANCE ERFECT CODES Wth SK-mtc Chat-5 w hav studd what ma th st ctt b calld: a -addtv - fct cds that s ths cds whch cct lss - addtv s ad m b -addtv fct cds that s ths cds whch cct lss ad s ad m CLASS-QUASI ERFECT CODES It ma b calld that classcal studs f cds a uas-fct cd s dfd as fllws: DEFINITION: QUASI ERFECT -ERROR CORRECTING CODE A uas fct - cctg cd ccts all s fw sts sm sts ad m tha sts Wth s csdd tms f classs f a SK-att ad csut mtc a uas fct - cctg cd ca b dfd sval was Csdg t a -addtv - a uas fct cds - -addtv ca b csdd fllwg dfft cass: 59

7 6 DEFINITION: -ADDITIVE CLASS QUASI ERFECT CODE A cd that ccts all -addtv lss s ad th mag l -addtv sts s a -addtv class uas fct cd Ths cds f th st f gv ad th att wll satsf th cdt < (4) DEFINITION: -ADDITIVE ~ CLASS QUASI ERFECT CODE A cd that ccts all -addtv lss s ad th mag s a all lss sts wth ts fm th tha lmts f s a -addtv ~ class uas fct cd Th cd gvs a cmbd st f S dgts: < S S S (4) () O ca csd fllwg futh sub-cass f (4): If all th ths a fm th S ad th th lft-had ss (4) b th fllwg: (43) () Nw cas (b) f all th ths a a t th tha fm th S ad th th lft-had ss (4) s th fllwg: (44) () Yt dfftl f all th ths a such that at last s fm whl ths a fm ad th th lft-had ss (4) b th fllwg:

8 ( ) ( ) (45) Fm th abv dscuss t ca b s that ths aach th ca b v ma dfft classs f th uas-fct cds asg fm a udlg SK-att f Z f cmg t a bud bablt w call tw thms gv blw vd b us chat f s csdd abv THEOREM A: Gv a SK-att { } m f Z m a cssa cdt f stc f a ( ) cd v Z cctg s -sts ach f wght s gv b t t t lg t t t THEOREM : Gv a SK-att { } cdt f th stc f a ( ) cd v m f Z m a cssa Z cctg lss sts s that ths sts a t j wh s cvd as a lmt f s gv b j j ( ) t 44 LOWER OUND ON THE ROAILITY OF ERROR FOR SK- METRIC CHANNEL I ths sct w ta u studs f -bablts ud addtv SK-att Class s ad latd buds 6

9 TWO CATEGORIES OF ERROR-ATTERNS Csdg cmmucat v th SK-Smmtc chal dfd abv w csd s th fllwg tw catgs ad th bablts: R Numb f -cctabl atts wth s lss sts wth ts asg b addt f dgts th tha ths f R Numb f -cctabl atts wth s m sts asg b addt f ts fm a f s ( R ) bablt f -cctabl atts wth s lss sts wth ts asg b addt f dgts th tha ths f bablt f -cctabl atts wth s m v R sts asg b addt f ts fm a f s THEOREM 4: Gv a SK-att { } m f Z m bablt f tasmttg v SC a cd caabl f cctg -addtv s cvg vcts that hav s fw sts wth addt ths sts f dgts th tha ths f s ( ( ) ) wh ROOF: W bg b cutg th -atts f th catg R that a -z ad - addtv lss sts ad th csd csdg bablt f th bg cvd v th chal Csdg a vct wth -z t th tha ths f wll hav t f Lt E dt th umb f -vcts that hav -z t th tha ths fm th 6

10 63 E Nt th bablt f cvg ths E vcts s wh s th bablt f cct tasmss f a dgt ad s th bablt f chag th tha addt f that f fm Smlal w ca fd E 3 E E ad csdg bablts f th cvg F ths s sts gal w hav W ca wth a vw t bta cls ss sus ad th S th ttal bablt f cvg vcts f ths catg th s

11 Wh ( a b) stads f < tms th bmal as ( a b) THEOREM 4 : Gv a SK-att { } f Z m bablt f m tasmttg v SC a cd caabl f cctg -addtv s cvg vcts wth addt ths sts f dgts fm a f s s m wh m ROOF: W stat wth fdg sss f bablt f cvg vcts wth -z ts Ths f s gv b th fllwg ss ( ) ( m m ) F a gal valu f ths ca b ut as m Th ttal bablt f cvg vcts f ths catg f w hav th bablt f gv b m COROLLARY 4: Gv a SK-att { } f Z m bablt m f tasmttg v SC a cd caabl f cctg -addtv s cvg vcts that hav s m sts wth addt ths sts f dgts fm a z t at last 64

12 65 ROOF: I gal w ma assum that > m- th > th m m m Th ttal bablt f cvg vcts f ths catg f w hav th bablt f gv b THEOREM 43: Gv a SK-att { } m f Z m f a cd caabl f cctg -addtv s sts bablt f tasmttg v SC s ) ( α wh α

13 66 ROOF: Eal f th cd ud csdat w hav csdd tw ts f - atts s th ttal umb f atts s R R R wth th csdg bablt f m F btag a bud th bablt f w csd that th cd s uas fct that s t s a cd caabl f cctg -addtv s fw sts ad ths ts a all -addtv sts ad m Th umb f cctabl atts u t sts s Ad th umb f th mag cctabl -vcts s α Als th umb f all th -cctabl atts that s ths utsd th uas fct csdat a sm f s sts ad th st ths havg a f th -z ts m sts Th umb f ths havg -z ts ad m sts a: It ma thus b s that R has vcts that -cctabl sm f whch a f s ad all ths f -z m sts F th uss f gttg a lw bud th bablt f f cvg R vcts cctg -addtv s tasmttg v SC s

14 67 R ) ( α Csdg dfft classs f atts tgth th bablt f th cd ud csdat has th lw bud gv b α Ths vs th sult HAMMING CASE Abv csdats wh ald t Hammg cas cma [78] ma that th class R wll b mt s that ) ( Ad w th bud tas th fm: α Th sult vd abv s scfc f addtv s ad th uas-fct csdat ta th It ca b tdd t th class addtv s ad stuats I th t thm w csd wh t s addtv THEOREM 44: Gv a SK-att { } m f Z m bablt f tasmttg v SC a cd caabl f cctg -addtv s cvg vcts that hav s fw sts wth addt ths sts f dgts th tha ths f ad s

15 68 Wh ad ROOF: W bg b cutg th s atts ad th csdg bablt f th cvg v th chal Csdg a vct wth -z t th tha ths fm ad wll hav - z t fm Th umb E s gv b Nt th bablt f cvg ths E vcts s Smlal w ca fd E 3 E E ad csdg bablts F ths s sts gal w hav Sus ad th

16 69 THEOREM 45: Gv a SK-att { } m f Z m f a cd caabl f cctg -addtv s sts bablt f tasmttg v SC s ) ( α wh α ROOF: Fm thm 4 ad 44 w hav R ) ( ad R m th ) ( m Th umb f cctabl atts u t sts s

17 7 Numb f th mag th cctabl s s α Als th umb f st f all atts t fallg f th uas fct cas ad a hgh sts s R ) ( α Th bablt f f a cd caabl f cctg -addtv s tasmttg v SC s ) ( α 45 CONCLUDING REMARK Mst algbac cds hav stuctud t cct scfc ts f s ut th chal bg bablstc atu th s a aa f labl f th cmmucat I ths chat w hav tducd a v galzd cct f addtv s tms f classs f SK-atts f Z Fm ths has fllwd th cct f Class-Addtv s ad adl Galzd SK-dstac fct Cds I ths aach atts hav als b dfd vaus dfft was ad s th studs that fllw fm t

18 Cdg s ud f dfft ts f chals Ou stud has tdd th wdl usd cct f a-smmtc Chals Csdg tw ts f -atts th a sults th bablts f s f th -a chals asg fm SK-atts Th th a buds bablt f s whch galzs aach ad sults bablt f s f a cd dsgd f dfft classs f s Wth galzat f addtv s ad f smmtc chal f th w sttg ud SK-studs th a sval th ssblts that as f futh studs 7