The MacWilliams Identity of the Linear Codes over the Ring F p +uf p +vf p +uvf p

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1 Reearch Joural of Aled Scece Eeer ad Techoloy (6): ISSN: 2-6 Maxwell Scefc Orazao 22 Submed: March Acceed: Arl 22 Publhed: Auu 5 22 The MacWllam Idey of he Lear ode over he R F +uf +vf +uvf aofa u ad Haq Ha School of Mahemac ad Phyc Hube Polyechc Uvery Huah 5 ha Abrac: I h udy a ew comlee weh eumeraor ad a ew ymmerzed weh eumeraor over he r R = F +uf +vf +uvf are defed. By u a ecal varable he comlee weh MacWllam dey ad he ymmerzed weh MacWllam dey over he r R are ve. Keyword: omlee weh eumeraor lear code MacWllam dey ymmerzed weh eumeraor INTRODUTION ode over r have bee uded exevely he a decade wh he emerece of he roud-break work doe Hammo e al. (99) whch hey looked a lear code over Z. Sce he may dffere ye of r have bee uded coeco wh cod heory. I 99 he weh drbuo of he lear code over r were uded (Wa 99). The eeralzed MacWllam dee of he lear code over he fe feld wa ve (Shromoo 996). The MacWllam dee of he lear code over he r F +uf were uded L ad he (2). Recely lear code over he r F 2 +uf 2 +vf 2 +uvf 2 were uded Yldz ad Karadez (2) ad cyclc code over he r F 2 +uf 2 +vf 2 +uvf 2 were uded Yldz ad Karadez (2) where he r F 2 +uf 2 +vf 2 +uvf 2 o a fe cha r. I h udy we udy he cod heory over he r R = F + uf + vf + uvf. A ew comlee weh eumeraor ad a ew ymmerzed weh eumeraor over he r R are defed. By u a ecal varable a ew comlee weh MacWllam dey ad a ew ymmerzed weh MacWllam dey over he r R are ve. LINEAR ODE OVER THE RING R The r R = F + uf + vf + uvf defed a a characerc r ubjec o he rerco u 2 = v 2 = ad uv = vu where rme. I eay o oberve ha R a local Frobeu r whch o fe cha or rcal. The deal ca be decrbed a: where I = fi uv fi u I v I u+v fi u v fi = R () I uv = uv(f +uf +vf +uvf ) I u = u(f +uf +vf +uvf ) = uf +uvf I v = v(f +uf +vf +uvf ) = vf +uvf I u+v = (u+v)r = (u+v)f +uvf I u v = uf +vf +vf Le R * = R!I uv we ca ee ha R* co of all u R I u v he uque maxmal deal ad ha o a rcal deal. I u v co of all zero dvor R. Defo : A lear code over he r R of leh a R-ubmodule of R. The comlee weh MacWllam dey of he lear code over he r R: Le R = { 2... } ome order. For examle = 2 =...ad o o. œx = (x x 2... x ) y = (y y 2... y )R he er roduc of x y defed a he follow: <x y> = x y +x 2 y 2 + +x y I h udy defe z = {x <x y> = œy} o be he dual code of. Defo : The comlee weh eumeraor of he lear code over he r R defed a: cwe c c c where c c he umber of aearace of he vecor c. Lemma 2: Th udy roduce a abrac whoe exoe wll be eleme of R uch ha 2 a b a b uv e where a br he I = for all ozero deal I of R: Proof: Th ca be how by rahforward calculao. For examle le = au+buv I u he: aubuv au buv au uv b ( ) I af bf af bf af b u b 2 2 au au e e 2 a F b a F e orreod Auhor: aofa u School of Mahemac ad Phyc Hube Polyechc Uvery Huah 5 ha 28

2 Re. J. Al. Sc. E. Techol. (6): Theorem : Le be he lear code of leh over R ad le z be dual. The wh ad a defed above we have: cwe 2... c cwe Proof: For ay he c c le: c x x Fc xr c x Fc xr c xr x c c x x Now uoe for fxed x R we coder he fuco f x from o R. f x defed a fx cc x. By he rucure of he er roduc we kow ha f x a R- module homomorhm. The by he defo of he dual code we have: ker fx c x c The for ay x we have x cx. Now uoe ha x h mle ha ker fx. By he roery of he homomorhm we kow ha Im(f x ) a ozero ub-module of R ad hece a ozero deal of R. The by Lemma 2 we have ha: whe x o x cx. Th mea ha: c Fc c x cwe... 2 whch equvale o ay ha: cwe Fc 2... c c (2) O he oher had le *(x y) deoe he Kroecker x y Dela fuco: *(x y) =. So: x y cx x Fc xr x x2... x R cj. j j j. j c j cx j j xj By he defo of he comlee weh eumeraor dey we have: Fc cwe c () omb (2) ad () we kow ha he heorem ca be roved.. Lemma : For all eleme R we have. R Proof: Smlar o he roof of lemma Yldz ad Karadez (2). Theorem 5: Le be a lear code of leh over R: he. Proof: By he defo of he comlee weh eumeraor dey U he heorem we have: cw... cwe... cwe cwe The by he lemma we e: cwe... cwe... Thu we have roved he heorem. 29

3 Re. J. Al. Sc. E. Techol. (6): THE SYMMETRIZED WEIGHT MAWILLIAMS IDENTITY OF THE LINEAR ODES OVER THE RING R Defo : lafy he eleme of R o 8 ube a D = {} D = I uv \{} D 2 = I u+v \I uv D = I v \I uv D = I u \I uv D 5 = {x = bu+cv+du b c F \ {} df b c (b+c)/ (mod )} D 6 = {x = bu+ cv+ duv b c F \ {}df b c (b+c) (mod )} D = R\I u v. Fuco I(@) defed a: I(a) = where ad ( =...). Defo 2: The ymmerzed weh eumeraor of he lear code over he r R defed a: we... cwe... I I 2 I Lemma : Wh he ame oao a above he whe we kow: D = D =! D 2 = D = (!) D = D 5 = (!) D 6 = (!)(!) D = (!) D D D Proof: By he defo of D ( = 2 ) we kow ha eay o be roved I eay o kow ha D D. By he lemma 2 we have D I D uv whe D 2 le = b(u+v)+duv where bf \ {} df we have: D2 b( uv) duv bfp \{} df bu ( v) duv bf\{} df Oher are mlar o be roved. Lemma : Wh he ame oao a above whe >2 we kow:. D If D he D... If D he D D D If D 2 he D D.. D 2 D 5 D 6 D If D he D D D D D 6 D If D he D D D. D D 6 D. If D 5 he D D 2 D 5 D 6 D If D 6 he D D D 2 5 D D 6 If D he D D 256 D Proof: We ju choce ) ad 8) o rove oher are mlar o be roved. If D le = cv+duv where cf \{} df he. D D D D whe D 2 le = b (u+v)+ d uv d F b F \{} he: D2 bf \{} df bf \{} D whe D le = b v+ d uv d F b F \{} he: D bf \{} df whe D le = b u+ d uv d F b F \{} he: D bf \{} bf bf \{} D 28

4 Re. J. Al. Sc. E. Techol. (6): whe D 5 le = b u+ c v+ d uv d F b F \ {} c F \{} b c ( b + c ) = (mod ) he:. D5 bf / df bf / D whe D 6 le = b u+ c v+ d uv d F b F \ {} c F \{} b c ( b + c ) (mod ) he: D6 bf \{} cf \{} cbdf whe D le = a + b u+ c v+ d uv a F \{} b c. d F he:. acv D af \ bf cf df 2. D If D D ( = 2... ) becaue a u we kow: {. D } = D The By lemma 8) D D eay o be roved. Theorem 5: Le be he lear code of leh over R he where we c we Y Y Y Y = +(!) +(!) 2 + (!) +(!) +(!) 5 + (!)(!) 6 + (!) Y = +(!) +(!) 2 + (!) +(!) +(!) 5 + (!)(!) 6! Y 2 = +(!)! 2!! +(!) 5!(!) 6 Y = +(!)! 2 +(!)! P! 5!(!) 6 Y = +(!)! 2! + (!)! 5!(!) 6 Y 5 = +(!) +(!) 2!!! 5!(!) 6 ; Y 6 = +(!)! 2!!! Y =! Proof: By he defo of he ymmerzed weh eumeraor dey ad he heorem we have: we... cwe... I I I 2 cwe. I 2.. I... I cwe. D D D By he lemma we kow: we we Y Y Y Y D j 2... j D Thu we have roved he heorem. j ONLUSION I h udy we uded wo kd of MacWllam dee of he lear code over he r R. Aoher dreco for reearch h oc of coure he coacyclc code ad he dual code over he r R. AKNOWLEDGMENT Th udy uored by Naoal Naural Scece Foudao of Hube Polyechc Uvery of ha (yjzq). REFERENES Hammo A.R. P.V. Kumar A.R. alderbak N.J.A. Sloae ad P. Sole 99. The Z leary of Kerdock Prearaa Gehal ad relaed code. IEEE T. Iform. Theor. :

5 Re. J. Al. Sc. E. Techol. (6): L Y. ad L. he 2. The MacWllam dey of he lear code over he r F +uf P. Aca Scearum Nauralum Uvera NaKae ha (2): 8-8. Shromoo K A ew MacWllam ye dey for lear code. Hokkado Mah. J. 25: Wa Z. 99. Quaerary ode. World Scefc Publcao o. Saore : 25-. Yldz B. ad S. Karadez 2. Lear code over F 2 +uf 2 +vf 2 +uvf 2. De.ode ry. 5: 6. Yldz B. ad S. Karadez 2. yclc code over F 2 +uf 2 +vf 2 +uvf 2. De. ode ry. 58: