
hamming (7,4)

coding siddhaantamlo, hamming (7,4) anedi oka sarala dosha-diddubaatu code, idhi moodu parity bitlanu jodinchadam dwara naalugu bitla daetaanu edu bitlugaa encode chestundi. idhi pedda kutumbamloni hamming kodla sabhyudu, conny "hamming code" anae padam tarachugaa eee nirdishta kodnu suuchistumdi. 1950loe pravesapettina hamming, aaa samayamlo, hamming bell telephone prayogasaalalo panichesindi mariyu error-prone punch card readertho visugu chendindi, anduke athanu error-correcting kodlapai panicheyadam praarambhinchaadu.
hamming code sandesamloni prati naalugu dayta bitlaku moodu adanapu check bitlanu jodistundi. hamming (7,4) algorithm edainaa single-bitt errarnu saricheyagaladu ledhaa anni single-bitt mariyu too-bitt errarlanu gurtinchagaladu. maroe maatalo cheppalante, edainaa rendu saraina sanketapadaala madhya kaneesa hamming dhooram 3, mariyu sweekarinchina padaalu pampinavaaru prasaram chesina sankethapadhaaniki garishtamgaa okadaaniki dhooramlo unte vaatini sarigga decode cheyavachu. dheeni ardham paeludu lopaalu sambhavinchani prasaara madhyama paristhitulaku, hamming (7,4) code prabhaavavantamgaa untundi (endukante edu bitlalo rendintiki madhyamam chaalaa dhwaninchedigaa undaali).
quantum samacharamlo, hamming (7,4) nuu stein code kosam besga upayogistaaru, idhi quantum error diddubaatu kosam upayoginche oka rakamaina css code. 
lakshyam.
hamming kodla lakshyam emitante, dayta bitlo single-bitt error "ledhaa" parity bitt "nuu gurtinchi saricheyadaaniki veelugaa ativyaapti chendutunna parity bitla samitini srushtinchadam. bahula ativyaaptini srushtinchagaliginappatiki, saadhaarana paddhatini hamming kodlalo pradarsistaaru.
encode chesina padamlo bitsnu prasaram chese yea parity bits cover chestayo eee pattika vivaristundi. udaaharanaku, "p" 2 bits 2,3,6 mariyu 7 laku samaana samaanatanu andistundi. idhi kaalamnu chadavadam dwara yea samaanata bitt dwara prasaram cheyabadina bitt cover cheyabadindo kooda vivaristundi. udaaharanaku, "d" 1 anedi "p" 1 mariyu "p" 2 thoo kappabadi untundi, conny "p" 3 kaadu, eee pattika tadupari vibhaagamlo parity-check matricks (h) thoo adbhutamaina polikanu kaligi untundi.
inca, paina perkonna pattikalooni samanatva niluvu varusalanu tolaginchinatlayithe
appudu diguva code generator matricks (zii) yokka 1,2 mariyu 4 varusalatho poolika kooda spashtamgaa kanipistundi.
kabatti, samanatva bitt kavarejeeni sarigga enchukovadam dwara, 1 yokka hamming dhooramtho unna anni lopaalanu gurtinchavacchu mariyu sarididdavacchu, idhi hamming kodnu upayoginchadam yokka point.
matricksnu hamming cheyadam.
hamming kodlanu matricks dwara leanier beejaganitam paramgaa lekkinchavacchu endukante hamming kodlu leanier kodlu. hamming kodla prayojanaala kosam, rendu hamming maatrikalanu nirvachinchavacchuh code generator matricks g mariyu parity-check matricks h:
paina paerkonnatlugaa, g yokka 1,2 mariyu 4 varusalu dayta bitsnu vaati samanatva bits kuu mapp chestunnappudu suparichitamgaa kanipinchaalih
migilina varusalu (3,5,6,7) daetaanu encode chesina roopamlo vaati sthaanaaniki mapp chestaayi mariyu aaa varusalo 1 matrame untundi kabatti idhi okelanti coppy. vaastavaaniki, eee naalugu varusalu saralamgaa swatantramgaa untaayi mariyu gurtimpu maatrikanu yerparustaayi (design dwara, yaadruchikam kaadu).
paina paerkonnatlugaa, hetch yokka moodu varusalu suparichitamgaa undaali. eee varusalanu sweekarinche chivaralo syndrome vektarnu lekkinchadaaniki upayogistaaru mariyu syndrome vector shoonya vector (anni sunnalu) ayithe sweekarinchina padam dosharahitam; sunna kaanidi ayithe, yea bitt tippabadindo viluva suuchistumdi.
naalugu dayta bits-vector p gaa sameekarinchabadataayi-g (i) dwara mundugaa guninchabadutundi.ii., gp) mariyu prasaram cheyabadina encode viluvanu ivvadaaniki maadyulo 2 nuu teesukunnaru. asalu 4 dayta bits edu bitsga maarchabadataayi (anduvalla paeru "hamming (7,4)"), paina perkonna dayta bitt kavarejeelanu upayoginchi samaana samaanatvaanni nirdhaarinchadaaniki moodu samanatva bits jodinchabadataayi. paina perkonna modati pattika prati dayta mariyu samaanatvam bitt madhya mappingnu daani tudi bitt sthaanaaniki (1 nundi 7) chooputundi, ayithe dheenini wenn rekhaachitramlo kooda pradarsinchavacchu. eee vyaasamlooni modati rekhaachitram moodu vruttaalanu chooputundi (prati samanatva bitt kosam okati) mariyu prati samanatva bitt cover chese dayta bitsnu kaligi untundi. rendava rekhaachitram (kudivaipu choopabadindi) okela untundi, conny, badulugaa, bitt sthaanaalu gurtinchabaddaayi.
eee vibhagam yokka migilina bhaagaaniki, eee krindi 4 bits (kaalam vectorga choopabadindi) nadustunna udaaharanagaa upayoginchabadataayih
chaanal coding.
manam eee daetaanu (coque _ 1) dhwaninche communication chhaanel dwara prasaram cheyalanukuntunnamu anukundam. pratyaekamgaa, binary cemetric chhaanel antey error avineeti sunna ledhaa okadaaniki anukuulamgaa undadu (idhi errarlanu kaliginchadamlo cemetric). inca, anni muula vahakalu samardhavantamgaa untaayani bhaavistaaru. prasaram cheyabadina code vard x nii nirnayinchadaaniki, memu g mariyu p yokka utpattini, modyulo 2 entreelatho teeskuntamuh
dheeni ardham, coque _ 1 nii prasaram cheyadaaniki badulugaa coque _ 2 prasaram cheyabadutundi.
gunakaaram gurinchi aandolana chendutunna prograamarlu, phalitam yokka prati varusa, varusa mariyu niluvu varusalu bitwaisga undatam mariyu guninchadam kante kalisi undatam valla erpade sett bits yokka janabha gananalo athi takkuva mukhyamaina bitt anie gamaninchaali.
prakkane unna rekhaachitramlo, encode cheyabadina padam yokka edu bits vaati sambandhita pradaesaalaloo choppinchabadataayi; tanikhi nundi erupu, aakupaccha mariyu neelam vruttaala samaanatvam samanamga undani spashtamavutundih
prasaara samayamlo, oka bitt tippabadithe, rendu ledhaa moodu vruttaala samaanatvam tappu avutundi mariyu eee moodu vruttaala samaanatvam samanamga undaalani telusukovadam dwara porapaatu ayina bitnu (samaanata bits loe okati ayina) nirnayinchavacchu.
samanatva tanikhi.
prasaram samayamlo etuvanti lopam sambhavinchakapothe, appudu sweekarinchina code vard r prasaram cheyabadina code vard x kii samanamga untundih
syndrome vector z nii pondadaaniki receiver h mariyu r nuu gunistundi, idhi lopam sambhavinchindo ledho suuchistumdi, mariyu alaa ayithe, yea code vard bitt kosam. eee gunakaaram cheyadam (mallii, maadyulo 2 loe namodulu):
syndrome z anedi shoonya vahakamu kabatti, graheeta etuvanti lopam sambhavinchaledani nirdhaarinchavacchu. dayta vektarnu g thoo guninchinappudu, aadharam yokka maarpu vector subspacega sambhavistundi, idhi h yokka kernell anae pariseelanapai eee mugimpu aadharapadi untundi. prasaram samayamlo aemee jaraganinta varaku, r anedi h yokka kernallo untundi mariyu gunakaaram valla shoonya vector labhistundi.
dosha savarana.
lekapothe, manam vraayagalamani anukundam
maadyulo 2, ikkada e "i" anedi suutram _ 7 unit vector, antey, suutram _ 8loe 1thoo sunna vector, 1 nundi lekkinchabadutundi.
eee vidhamgaa paina perkonna vyakteekarana suutram _ 8 sthaanamlo oke bitt lopaanni suuchistumdi.
ippudu, manam eee vektarnu h thoo gunisteh
x anedi prasaram cheyabadina dayta kabatti, idhi lopam lekunda untundi, phalitamgaa, h mariyu x yokka utpatti sunna avutundi. eee vidhamgaa
ippudu, suutram _ 8 praamaanika aadhara vector thoo h yokka utpatti h yokka aaa niluvu varusanu enchukuntundi, eee h yokka niluvu varusa sambhavinche pradaesamloe lopam sambhavistundani manaku telusu.
udaaharanaku, manam bitt #5 loe oka bitt error nii pravesapettaamu anukundam.
kudivaipuna unna rekhaachitram erra mariyu aakupaccha vruttaalalo bitt error (neelam vachanamlo choopabadindi) mariyu chedu samaanatvam (erupu vachanamlo choopabadindi) nuu chooputundi. erupu, aakupaccha mariyu neelam vruttaala samaanatvaanni lekkinchadam dwara bitt errarnu gurtinchavacchu. chedu samaanatvam kanugonabadithe, appudu chedu samanatva vruttaalanu matrame ativyaapti chese dayta bitt anedi lopamtho bitt avutundi. paina perkonna udaaharanalo, erupu mariyu aakupaccha vruttaalu chedu samaanatvaanni kaligi untaayi, kabatti erupu mariyu aakupaccha yokka kuudaliki anugunamgaa unde bitt, conny neelam kaadu, tappu bitnu suuchistumdi.
ippudu,
idhi h yokka aidava niluvu varusaku anugunamgaa untundi. inca, upayoginchina saadhaarana algorithm ("hamming code#general algorithm chudandi") daani nirmaanamloo uddeshapoorvakamgaa undhi, tadwara 101 yokka syndrome 5 yokka binary viluvaku anugunamgaa untundi, idhi aidava bitt paadaindani suuchistumdi. eee vidhamgaa, bitt 5loe oka lopam kanugonabadindi, mariyu sarididdavacchu (daani viluvanu tippandi ledhaa tiraskarinchandi):
idhi sarididdabadina andukunna viluva, ippudu, paie nundi prasaram cheyabadina viluva x thoo saripolutundi.
decoding.
okasari andukunna vector lopam laenidigaa nirnayinchabadina tarvaata ledhaa lopam sambhavinchinatlayithe (sunna ledhaa oka bitt lopaalu matrame saadhyamani bhaavinchi) sarididdabadina tarvaata andukunna daetaanu asalu naalugu bitlaloki tirigi decode cheyalsina avasaram undhi.
modata, oka maatrika r nii nirvachinchandih
appudu andukunna viluva, pr, rr kii samanam. paie nundi nadustunna udaaharananu upayoginchadam
aneka bitt lopaalu.
eee pathakaanni upayoginchi oke bitt lopaalanu matrame sarididdavachani choopinchadam kashtam kaadu. pratyamnaayamgaa, doshaalu sambhavinchinappudalla h yokka utpatti sunna kaanidani gamaninchadam dwara, oke mariyu double bitt lopaalanu gurtinchadaaniki hamming kodlanu upayoginchavacchu. prakkane unna rekhaachitramlo, bits 4 mariyu 5 tippabaddaayi. idhi chellani samaanatvamto oka vruttam (aakupaccha) matrame istundi, conny lopaalu tirigi pondalevu.
ayithe, hamming (7,4) mariyu ilanti hamming sankethaalu single-bitt lopaalu mariyu too-bitt lopaala madhya thedaanu gurtinchalevu. antey, rendu-bitt lopaalu oka-bitt lopaala maadirigaane kanipistaayi. rendu-bitt lopam vishayamlo dosha diddubaatu chesthe phalitam tappu avutundi.
adevidhamgaa, hamming sankethaalu ekapaksha moodu-bitt doshaanni gurtinchalevu ledhaa tirigi pondalevu; rekhaachitraanni pariganinchandih aakupaccha vruttamlo bitt (rangu erupu) 1 ayithe, samanatva tanikhi shoonya vektarnu tirigi istundi, idhi sanketapadamlo lopam ledani suuchistumdi.
anni sanketapadaalu.
muulam kevalam 4 bits matrame kabatti, appudu prasaram cheyagala 16 padaalu matrame unnaayi. adanapu samaanatvam bitt upayoginchinatlayithe enimidi-bitt viluva cherchabadutundi ("adanapu samaanatvam bitt thoo hamming (7,4) code chudandi"). (dayta bits neelam rangulo chuupabaddaayi; parity bits erupu rangulo chuupabaddaayi; mariyu adanapu parity bits aakupaccha rangulo chuupabaddaayi.)
e7 jaalakam.
hamming (7,4) code e7 jaalakaaniki daggari sambandham kaligi untundi mariyu vaastavaaniki, daanini nirminchadaaniki upayoginchavacchu, ledhaa marinta khachitamgaa, daani dwandwa jaalaka e7 ≤ (e7 kosam idhe vidhamaina nirmaanam dwandwa kodnu upayogistundi [7,3,4] 2). mukhyamgaa, anni vahakalu "x" nii z "7" loe "x" thoo kalisi (maadyulo 2) hamming (7,4) yokka code vard kii teesukoni, 1/dwara reselling cheyadam dwara, jaalakaaniki e7 ≤ istundi.
idhi jaalakaalu mariyu sankethaalu madhya marinta saadhaarana sambandhaaniki oka pratyeka udaaharana. udaaharanaku, samaanatvam bitnu jodinchadam dwara utpannamayye podiginchina (8,4)-hamming code kooda e8 laatistoe sambandham kaligi untundi. 

