Partition and the Perfect Codes in the Additive Channel

Transcription

1 Oe Joual of Dcete Matheatc htt://dxdoog/36/od333 Publhed Ole July 3 (htt://wwwcog/oual/od) Patto ad the Pefect ode the Addtve hael Gab Movya BI Gou Mocow Rua Eal: gab@fbtu Receved Mach 33; eved May 3; acceted May 7 3 oyght 3 Gab Movya h a oe acce atcle dtbuted ude the eatve oo Attbuto Lcee whch et uetcted ue dtbuto ad eoducto ay edu ovded the ogal wo oely cted ABSRA May oble of dcete otzato ae coected wth atto of the -deoal ace to ceta ubet ad the equeet eeded fo thee ubet ca be geoetcal fo tace the hecty o they ca be coected wth ceta etc fo tace the equeet that ubet ae Dchlet ego wth Hag etc [] Ofte atto to oe ubet ae codeed o whch a fuctoal otzed [] I the eet wo the atto of the -deoal ace to ubet wth zeo ltato ae codeed Such atto allow u to cotuct the et of the gou code V ad the et of the chael A betwee the abtay eleet V ad A havg coectg elato betwee the Decto of oe clae of both efect ad efect code the addtve chael ae eeted too A way of cotuctg of gou code coectg the eo the addtve chael eeted ad th ethod a futhe geealzato of Hag ethod of code cotucto Keywod: Patto; Pefect ode; Addtve hael Itoducto Let В be a Galua feld of two eleet ad B be a lea vecto ace o that feld We code the faly of the ubet atfyg the followg codto fo all : ) B ; ) ; () 3) ; ) x (uato wth eect to ) od x he ft thee oete ae uual fo atto of the ubet ad the lat o-zeo oe eflect the ecfcty of the futhe uage fo cotuctg of the coectg code he cae B () atculaly otat becaue t lead to cotuctg of the efect code Below the te atto of the et ued the ee of () e t the atto to zeo ubet he oble of extece cotuctg ad atto of fo the gve ad have ot oly cobatoal-et teet but alo that coecto wth coectg code cotucto It wothy to ote that coectg code theoy the decodg ego fo atto of the ace B f decodg ego a do ot ovela each othe oequetly oe code clae atculaly the efect code the addtve chael ae t oble to cotuct the atto Below the exale wth we leave out the ubet whch the zeo vecto Exale the atto of B f: 3

2 G MOVSISYAN 3 Exale the atto of the ace 3 5 B f: 3 5 Exale the atto of the ace B f: 3 3 It ee fo the above exale that the ace B ca be attoed ay way wth eect both to the ube ad the owe of the ubet Fo the atto of the et oe ca obta the atto tag the ubet away fo We eet (wthout oof) the followg lea that decbe oe tval oete of the atto Lea Fo evey z z z z the followg tae lace: a) z z ; b) z z ; z \ z c) z z ; z \ z d) z z \ z\ z If fo we wll tae he we code the atto tag to accout that the eceay codto of the extece the evee of the ube f odd he followg cotuct of the dect oduct allow buldg ew atto out of the gve oe: Lea If ad ae the at- to of the et В ad В eectvely the thee ae atto of the ubet В Poof Let: whee: y y y x x x Let u eeet the dect oduct-et fo of the atx: x y x y x y x y the (3) Fo evey a We defe the et the followg way: a) Fo evey the et cota oly oe eleet of ay le ad ay colu of atx (3) ad t atfe the followg co- dto: o a of all the et ovelaed ad evey oe of thee et ha the owe ; that : b) ; c) Let u code the et: Fo defto of ad f ad thoe of f we have: x (uato wth eect to od ) x ad a: he a atto of the ace

3 G MOVSISYAN В heoe If a dvo of ad a dvo of fo ay otve tege the ad ae atto of the ace B Poof It follow fo the theoe codto that Let u aly ducto ethod wth eect to Fo т we eet B the fo B B \ whee he we have the tval atto of B Let u aue that fo thee the atto of the ace B Alyg Lea wth eect to the atto of B we ob- of B ad ta the atto fo В oe- quetly thee ext the atto of B whee We code whee ad defed the followg way: l l l It eay to ove that atfe codto () ad () ad coequetly a atto of B QED Now we ove the extece of the atto whee he tateet hold tue fo Let u aue that the tateet hold tue fo all a well ad ove t fo We eet В В В whee В x x x x A a tege ad the a tege oequetly alo a accodg to the auto thee ext the followg atto of the ace B : tege A We code whee x B x B A: fo x ad x fo all the t eough to ove that We wte: x B x B hat a atto of B QED Now we ae gog to decbe the cotucto of the gou code et algoth ad that of the chael et ug the atto of the et fo the ND ace to zeo ubet It oved that ay code of the cotucted et coect all eo of evey addtve chael the et of the eectve chael A addtve chael gve by the et of vecto of eo A y y y B ; ay vecto y at the ext of uch a chael ha the fo: y x y whee x В the tal vecto y A ad the addto oeato wth eect to od [3] he eghbouhood of the ode of t of the vecto х B wth eect to B defed the followg fo []: t t х x y x х y х x t A х tato: t t х хb doe ot deed o x we ue the deo- he code V coect the eo of the addtve chael A y y y f the followg codto ae ovded: А v А v whee v v V v v lacal boudae of Hag ad Vahaov- Glbet fo the owe of the code V coectg the eo of the addtve chael A have the followg fo [5]: V A A he a ta fo the gve chael A the cotucto of the axu volue code coectg the eo of the chael A he code V B called efect fo the addtve chael A B f the followg codto atfed:

4 G MOVSISYAN 5 V () A he code V B called qua-efect fo the chael A B f fo ay оf A \ A the code V efect fo the chael A I othe wod the qua-efect code V fo the chael A atfe the codto: A v A v v v V v v ; ) ) A v B whee vv A A We deote by l A the gou code fo B of the ode l coectg all the eo of the addtve chael A We def e the oduct of the Boolea atx Н а o the deo т ad the vecto x xx x the followg way: Нx zz z whee z ax (uato wth eect to od ) Ay () atx H havg the deo l called checg fo the code l A f fo all code vecto ad oly fo the the followg equalty tae lace: Нx whee all oeato ae caed out wth eect to od ([6]) o buld the code V coectg the eo of the addtve chael we ue the followg cotuct coected wth the atto eeted above Ft we buld the addtve chael the the gou code coectg the eo of that chael Let be the egatve tege ad thee be the et whee We code the atce a of the followg fo: Eесли а Eили E если \ Hee E the ut atx of the ode ad E the logc egato of E We buld the chael A B whee l l l cooed of the vecto x whee ad fo all le of the Boolea atx gve the followg way: a a (5) a Exale We buld a chael fo the cae: 5 3 ad 3 Ug the defto of the ube vecto x we obta: x x x x A a3 E E th e followg fo: A A ad the B 3 ; the the chael have

5 6 G MOVSISYAN NB he bloc а fo cotuctg the chael defed two way fo all ; coequetly the et A of uch chael ha the followg owe: Let р А f log whee othewe It obvou that cbed chael A ha the owe: Let ad the above de- A (6) be oe of the atto decbed above We tafo the faly the followg way: we tae fo each a vecto z ad thow t away eeg all othe vecto the foe fo We deo te the obtaed faly by whee: zf \ othewe NB he et deed o the choce of the vecto z fo ad the checg a tx z Т Т z Н z defe the code V z z z oe to oe; coequetly the et of the code V V z z z z ha the owe: V! We code the gou code V V B havg the checg atx H H ad the addtve chael А A fo We ove that the gou code V B havg H a t checg atx coect all eo of the cha el А A e V А o ove th t eough to how that fo ay xy А x y tae lace: x y V Let x xx x yyy y whee x y B fo all It eay to how that: x y Hece tag to accout that Т Т Т Н ha the deo: ad (7) the colu ube of the ub-atx wth thoe of atx (5) whee the bloc we obta: a y H x x y x y x y cocde located We have fo the defto of the chael A : a) hee ext a fo uch that fo all \ the vecto x zeo; b) hee ext a fo uch that fo all \ the vecto y zeo Hece we obta tag (8) to accout that thee ext a a fo whch: H x y x y (9) It follow fo the cotucto of the atx H that all the colu ae dffeet; coequetly fo ay vecto z V the equalty Нz tae lace f the z -weght of the Hag vecto z o e tha two х heefoe we code x y A fo whch x y he followg cae ae oble: a) he vecto x y ae the le of atx (5) he we obta fo (7): x y Hece tag (9) to accout we obta that thee ext uch vecto z z that: z z f x y Hx y z zf x y z \ z zf x y z \ z\ z We obta alyg Lea : Hx y e x y V b) Oly oe of the vecto x y a colu of atx (5) he we have fo (7): x y Hece tag (9) to accout we obta that thee ext the vecto z z uch that: H x y z f x y z zf x y z \ z z f x y z \ z zf x y z \ z\ z (8)

6 G MOVSISYAN 7 Alyg Lea we get: H x y e x y V c) Both vecto ae ot the le of atx (5) he we have fo (7): x y ag (9) to accout we get that thee ext uch vecto z z тthat: z f x y z \ z Hx y z f x y z \ z zf x y z \ z\ z Aga alyg Lea we get that fo ay vecto xy А x y tae lace: Н x y те coequetly V А x y V A a eult we have that evey co de V V coect the eo A of the et of ay chael A Futhe oe f a atto of B the fol lowg ta e lace: Hece we have tag (6) to accout that the code V А atfe the codto () that t efect I eult we get the followg tateet heoe If the evey gou code V V coect all eo of ay chael A A e V А oollay If a atto of В the evey gou code V V coect the eo of ay chael A A ad t efect NB 3 If the the above decbed ethod of buldg of gou code the Hag ethod of gou code coectg the eo of the cha el A ; Let u c hooe the above decbed algoth of cot uctg the et of chael tag to accout the followg codto: \ \ a odd ube We buld the et of chael A A the followg way; ay chael А A cooed of the vecto x whee l l l beg of all le of the Boolea a- tx gve the followg way: a a a Hee a a atx of deo havg the fo: Ef a Eothewe It obvou that the above decbed ocedue of cotuctg uquely defe the et A of the o whch: A zeo chael fo oequetly the followg hold tue oollay If a atto of the ace В the the efect code тv V coect the eo of the zeo chael A oollay 3 h e efect code V l А defe the atto: l l \ uquely A A x x V \ of the ace В f A the zeo chael Exale 5 We code the atto 3 3 whee: hoog z z we get the checg atx : oequetly the coeodg efect code: V z z coect the eo of the zeo chael A of the followg fo: I eult we get the code 5 V z z z A A

7 8 G MOVSISYAN fo xvz z z all A x 5 ae zeo et ad they atto the B we get the atto 78 A \ A x xv z z z \ 5 Now we ca get the followg efect code ad the atto fo the above atto a la way oequetly oollae ad ad 3 allow u to buld the equece of the atto of the ace ad the equece of the efect code a well Exale 6 We have fo Exale the atto of the ace B 3 Ug th atto fo we get the atx Т Т Т 3 : Whch the checg atx of the efect code А A ha the fo: 5 А whee the chael Exale 7 We ue the atto a the ecedg exa- 3 le ad we buld the 3 fo 3 z3 : 3 Т Т Т he we buld the atx 3 : whch the checg atx of the code А whee A oe of the chael the et A3 F o tace: oollay If a atto of the ace В ad fo oe tege l tae lace 3 l l the the gou code wth the checg atx Н H

8 G MOVSISYAN 9 -qua-efect Exale 8 We code the atto of the ace B of exale 3 Fo ad z we buld 3: 3 he we buld the atx Т Т Т 3 Whch the checg atx of the code А whee A oe of the chael the et A Fo tace: Now we ae gog to code the cae he teet th cae due to the followg ccutace Accodg to heoe extece of a atto B deed oly o the aaete ad ad th lfe the algoth of both code ad coucato chael decbed Bede the cae clafcato of buldg both code ad cha el - lfed a well It follow fo theoe ad theoe ( ) the followg heoe 3 If a dvo of ad a dvo of fo ay otve tege ad f A A the thee ext the efect code А fo ad fo whee he oof la to the oe fo heoe I the followg two exale we buld two dffeet chael ad the code coeodg to both ug the aaete Exale 9 Ug the atto 3 5 of the ace B we have fo exale two: 3 5

9 G MOVSISYAN Fo we get the checg atx Т Т Т 3 5 whch the checg atx fo the efect code 5 А whee the chael А A ha the followg fo: whch the checg atx fo the efect code 39 А de the chael А A 3 : We co Exale U g the atto of the ace B 3 5 we buld fo exale the fo the et z z z ad fo the vecto he we buld the atx : Т Т Т 3 5 At the ed of the eet ae we code the gou efect code bult though the atto fo р ad coectg the eo of the chael А A the ace B Exale We code the atto В \ fo ad В \ \ We tae the cae the checg atx of the gou efect code A A ha the followg fo: A whee

10 G MOVSISYAN Let u have a cloe lo o at the gou efect code А whee of coue fo 3 the cae that the 3 a atto of the ace B he fo the chael 3 A А ha the followg fo: ag th ad oollay to accout we get: he efect code A a qua-efect code the addtve chael: A f 3 3 A A Addeda Zeo Matce I th addeda we code the coecto betwee the zeo et ad the deadloc tet [7] F o the futhe dcuo t oe coveet to code the et of vecto a a atx havg thoe vecto a t le of the gve et Let F be the ace of the atce o Galua feld Defto he atx F called ull-atx f the u of t le a zeo vecto Moeove the a tx F called egula f all t colu ae dffeet It obvou that the egula atx coeod to the ubet of owe B Poble Decbe th e et P of egula atce Poble Fd the ube of the ull-atce Poble 3 Decbe the atto of B though the zeo ubet Exale ) If the the the egula ullatx ) If the thee doe t ext a egula ullatx 3) If 3 the the followg atce ae egula ull-atce: whee ay egula ull-atx of F 3 ) If the: o whee ay egula ull-atx F 5) Let 5 the the egula ull-atce ae a follow: ad whee a abtay ull-atx F 5 Now we code the followg et of atce F F F F : We toduce atal ode equg: If the atx ca be obtaed fo tag away oe et of colu Defto he atx the cla P called extee (o deadloc) f fo t follow that Exale 6) We decbe all extee atce P 5 : 3 Defto he two atce ad F ae called equvalet f ca be obtaed fo by euta to of le ad colu (th equvalecy deoted by: ~ ) Exale 7) he atce ad 3 exale 6 ae equvalet becaue 3 obtaed fo by the eutato of the d ad 3d colu

11 G MOVSISYAN I coecto of the above defto we ue the fol lowg codg of the atce of F We ueate all the colu of the legth choog fo tace the lexcogahcal ode Let the coeodg ueato be L L L he we ut a vecto of the legth to coeo de ce wth F We get L whee the ube of the colu equal to L the atx We call the vecto L colu vecto of the atx he the followg obvou: Stateet If the atce ad ae equvalet the L L Now we decbe all extee atce fo a fxed We deote the et of all uch atce by P he ele- et of the cla P have the followg oete: ) If P the h oety follow the fact that all the le of the atx ae dffeet ) If P the fo L we have: Ideed f? the thee ae detcal colu the atx But the we ca tae oe of the away ad the le the obtaed atx wll aga be dffe- et ad th ea that P 3) If P thee ethe egula o ut colu the atx h tateet oved la to the ecedg oe ) Each of the colu of the atx P ha eve ube of ut h follow the fact that the atx a ull-atx he gfcace of the toduced defto ad the above eult that they ae oble to obta ay atx ut addg oe ull-atx to P h due to the fact that tag away colu out of ay atx P lead to a atx P ocluo Obtag the atce P out of the atx P a oble of buldg deadloc tet fo the gve atx [7] REFERENES [] V K Leotyev G L Movya ad J G Magaya Patto of N-Deoal Sace o GF() to Dchlet Rego Vet R AU Vol No 6- [] G L Movya ad J G Magaya Otal Set the N-Deoal ube Uchee Za Vol 7 No [3] V K Leotyev G L Movya ad J G Magaya Pefect ode Addtve hael Reot of RAS Vol No [] V K Leotyev G L Movya ad J G Magaya oecto of Eo the Addtve h ael Vet RAU Vol No -5 [5] V K Leotyev G L Movya ad J G Magaya O Pefect ode Addtve hael Poble of Ifoato oucato Vol No 8-9 [6] F J M Wlla ad N J A Sloae he heoy of Eo-oectg ode Bell Laboatoe Maay Hll 977 [7] N A Solovyov et (heoy otucto Ue) Scece N ovob