Group Codes Define Over Dihedral Groups of Small Order

Transcription

1 Malaysan Journal of Mathmatcal Scncs 7(S): 0- (0) Spcal Issu: Th rd Intrnatonal Confrnc on Cryptology & Computr Scurty 0 (CRYPTOLOGY0) MALAYSIA JOURAL OF MATHEMATICAL SCIECES Journal hompag: Group Cods Dfn Ovr Dhdral Groups of Small Ordr * Dns C.K. Wong and Ang M.H Dpartmnt of Appld Mathmatcs and Actuaral Scnc, Faculty of Engnrng and Scnc, Unvrst Tunku Abdul Rahman Stapak, Off Jalan Gntng Klang, 500, Kuala Lumpur School of Mathmatcal Scncs, Unvrst Sans Malaysa, 00 Pulau Pnang, Malaysa Emal: dnswong@utar.du.my and mathamh@cs.usm.my *Corrspondng author ABSTRACT Th study of group cods as an dal n a group algbra has bn dvlopd long tm ago. If char(f) dos not dvds G, thn FG s smsmpl, and hnc dcomposs nto a drct sum FG = FG whr FG ar mnmal dals gnratd by th dmpotnt. Th dmpotnt provds som usful nformaton on dtrmnng th mnmum dstanc of group cods. In ths papr, w study dhdral group cods gnratd by lnar dmpotnts and nonlnar dmpotnts for dhdral groups of ordr,, 0 and. Our prmary task s to dtrmn th paramtrs of ths famls of group cods n ordr to obtan cods whch nar to attan th Snglton bound. Kywords: Group algbra, group cods, Snglton bound, lnar dmpotnts, nonlnar dmpotnts.. ITRODUCTIO Error corrcton or dtcton has bcom an mportant ssu wth th problm of rlabl communcaton ovr nosy channls. Snc thn group algbra cods hav bn a focus of ntrst n th mathmatcal communty n rlatng cods structurs by usng algbrac structurs. Group algbra cods gand ntrst aftr Brman showd n 97 that cyclc cods and

2 Dns C. K. Wong & Ang M. H Rd Mullr cods can b studd as dals n a group algbra FG, whr F s a fnt fld and G s consdrd, n ach cas, a fnt cyclc group and a -group rspctvly. On th othr hands, th frst nvstgatons of non- Ablan group algbra cod was don by F. J. Macwllams. Rcntly, P. Hurly and T. Hurly (Hurly (007)) construct group rng cods from zro dvsors and unt n group rngs n whch cas th cods dfnd may not b dal. In ths papr, w study cods dfnd ovr group algbra, whch s an dal. A group algbra cod n FG s dfnd as a on-sdd (lft or rght) dal n FG. If G s cyclc or Ablan, thn vry dal n FG s th cyclc or Ablan cod, rspctvly. Rfr (Brman (97) and (Brman (99)) for mor dtals on cyclc and ablan group cods, and (How and Dns (00)) for a class of nonablan group algbra cods. Th studs of group algbra cod n FG dpndd soldly on th chocs of F and G. In gnral, w can study group algbra cod n FG from th followng pont of vws: If gcd ( char( F ), G ) =, thn FG s smsmpl (rfr Thorm 5. (Isaacs, 997)), that s, FG s a drct sum of som mnmal dals, say s FG = I. Each I s gnratd by an dmpotnt =,.., I FG =. Lt E = { } s. Any dal I of FG s a drct sum of som of th I, = t say I = I k, t s. W say that I s gnratd by { } t. Lt = E \{ } t k k= k. k k = Thn I { u FG u 0 r r } = =. = For tchncal rason, w dnot I by I. ot that plays th rol of party chck matrx dfnng a lnar cod, and so w xpct to drv som nformaton about th mnmum dstanc of I from. Rcall som notaton and dfntons: Th lngth n of a group cod I b G. Th wght of any lmnt u FG s dfnd to = λ g s qual to g { λ λ 0 g g } and s dnotd by wt ( u ). If I has dmnson k (as a vctor spac ovr F ) and mnmum dstanc d = d ( I ) = mn{ wt ( u) 0 u I u}, thn I s calld an ( ),, n k d group cod. For mor nformaton on codng thory, plas rfr (Sloan and Macwllam, 97). 0 Malaysan Journal of Mathmatcal Scncs

3 Group Cods Dfn Ovr Dhdral Groups of Small Ordr In ths papr, w consdr group algbra cods dfnd ovr dhdral groups of ordr,, 0 and. Som bascs proprts of nonablan group cods wll b drvd n Scton, thn som proprts n dhdral group wll b drvd. Fnally, th mnmum dstanc of dhdral groups of ordr,, 0 and wll b studs n Scton and hnc som group algbra cods whch nar to attan th Snglton bound wll b obtand.. PRELIMIARY Most obcts n ths papr ar rprsntd n trm of group algbra FG. Th group algbra FG { a g a F g g } = s th fr F modul ovr a fnt group G whr G can b rgardd as an F bass for FG. Th addton and scalar multplcaton ar dfnd as follows. For any u = λ g, v = β g FG and F u + v = λ β g g + and g g g λ, ( ) λu = ( λλ ) g. Morovr, multplcaton n G nducs multplcaton n g FG as u. v = γ k whr γ = k k λ β. By ths opratons, FG s an g h k G gh= k G assocatv F algbra wth dntty = F G whr G and F ar th dntty lmnts of G and F, rspctvly. G can b vwd as contand n FG, and hnc th lmnts of G consttut th codng bass for cods vwd as subspacs of FG. W vw G as g n FG. Morovr, for ( ) A = a g FG, dfn A = a g. For mor nformaton on group g algbra, plas rfr (Passman (977)). g From now onward, w us th followng dfnton. Dfnton.. Lt G b a group and F b a fld such that gcd char F, G =. If E s th st of all dmpotnts of FG and E, ( ( ) ) thn th group cod gnratd by s I { u FG u 0 } = =. Proposton.. Th group algbra cods I dfnd n Dfnton. s a lnar cod ovr F. For any postv ntgr n, th dhdral group of ordr n can b n D = r s 0 n,0, r = s =, rs = sr. From rprsntd as { } now onward, all groups G ar D and all group algbra cods I ar n Malaysan Journal of Mathmatcal Scncs 0

4 Dns C. K. Wong & Ang M. H dfnd ovr D. Frst, to obtan th dmnson of I n, w nd th followng rsults. Thorm.. (Thorm.7, Jams and Lbck (99)). Lt K b a fnt group of ordr n, and F b an algbracally closd fld wth gcd char F, G =. Thn FK Mat ( F )... Mat ( F ), whr ( ( ) ) n n s = + +. FK has xactly s nonsomorphc rrducbl moduls, of n n... n s dmnsons n,..., n, and s s th numbr of conugacy classs of K. s = whr E s th st L Rmark.. Snc FG ( FG ) ( FG E ) E L conssts of all lnar dmpotnts n FG and E s th st conssts of all nonlnar dmpotnts n FG ; and furthrmor E = E E. ot that f L E, thn dm L ( FG ) = ; and f E, thn dm ( FG ) =. (Scton., Jams and Lbck (99)). Thrfor, f = whr E and E, thn L L L ( I ) ( FG) L dm = dm. As dm( FG) = G = n, thn dm( I ) n =. Th nxt thorm on th numbr of conugacy classs of found n (Scton.; Jams and Lbck (99)). L D can b Thorm.5. Th conugacy classs of D ar as follows: () If n s odd, thn D has ( n + ) conugacy classs: ( n ) / ( n ) / {},{ r, r },...,{ r, r },{ s, rs,..., r s}. () If n s vn and n = m, thn D has m + conugacy classs: m m ( m ) + {},{ r },{ r, r },...,{ r, r },{ r s :0 m },{ r s :0 m }. By usng Thorm.5 and rsults from (Chaptr, and 5; Jams and Lbck (99)), w obtan th followng proposton. ot that D ' dnot th commutator subgroup of D. 0 Malaysan Journal of Mathmatcal Scncs

5 Proposton.. Lt ntgr, thn (a) Irr ( D n ) Group Cods Dfn Ovr Dhdral Groups of Small Ordr D b th dhdral group of ordr n, whr n s any ( n + ), f n s prm, = ( n + ), f n = p, whr p s any prm. r, f n s prm, (b) D ' n = r, f n = p, whr p s a prm. (c) (d) D n D has n D ' lnar charactrs, whr D, f n s prm, n D ' = n, f n = p, whr p s a prm. D has ϖ non-lnar charactrs, whr n n, f n s prm, ϖ = n, f n = p, whr p s a prm. Proof. Part (a) s ust a drct consqunc from th fact that th numbr of rrducbl charactrs s qual to th numbr of conugacy classs. For part D (b), snc n Dn = and so r r s ablan, thn D ' r, rfr Thorm.0 n (Isaacs, 99). If n s prm, thn D ' = or D ' = r. If D ' =, thn D s ablan whch s mpossbl. Thrfor, w conclud n that D ' = r. xt, assum n = p, whr p s a prm. ot that D D r n n Snc r r D n = r r = and so = p, thn thr D ' = or n D ' r s ablan, thn D ' r. = r, and hnc th rsult wll follow drctly. Part (c) follows from part (b). Part (d) follows drctly from part (a) and (c). Q.E.D. Th followng lmma s usd to obtan th mnmum dstanc of I. Malaysan Journal of Mathmatcal Scncs 05

6 Lmma.7. If, thn I Proof. If all and so u I Dns C. K. Wong & Ang M. H I and so d ( I ) d ( I ). u I, thn u = 0 for all. Snc, thn u = 0 for. For th scond assrton, assum d ( I ) t wt u wt v, v I, thn d ( I ) t u I = t. If wth wt ( u) = and ( ) ( ) u I wth wt ( u) = t and wt ( u) wt ( v), for som v I othr hand, f thn d ( I ) Q.E.D. =. On th >, < t. Thus, th rsult follows drctly.. MIIMUM DISTACE OF DIHEDRAL GROUP CODES. Cods dfnd ovr FD Lt H = r r = and so D = H sh. From Proposton., D conssts of thr rrducbl charactrs (two ar lnar and on s nonlnar) χ, χ and χ, and ach of ths charactrs wll corrspond to a unqu dmpotnt (rfr Proposton.0; Jams and Lbck, 99) as follows: χ = ( H + sh ), ( H sh ) χ = and 0 Malaysan Journal of Mathmatcal Scncs χ = H. Lt u = λ + λ r + λ r + λ s + λ rs + λ r s b any lmnts n FD, λ F for 5 =,,,, 5,, thn u = λ () u ( ) = ( λ λ ) = = = u = + + r + + r + () ( ) ( ) ( ) λ λ λ λ λ λ λ λ λ ( λ λ λ ) s ( λ λ λ ) rs ( λ λ λ ) r s Lmma.. Lt, and b thos dmpotnts n FD as constructd abov, thn: d I = for =,. () ( ) () ( { } ) { } d I =. ()

7 Group Cods Dfn Ovr Dhdral Groups of Small Ordr Proof. W prov part () for th cas =. Th cas = can b provd n a smlar mannr. Assum u = λg I{ } for any g D and 0 λ F such that ( ) wt u =. By quaton (), u = λ 0. Hnc, w conclud that u = λg I and so d I. Clarly, u = g h I { } { } for any dstnct { } ( ) g, h D bcaus by quaton (), u = ( ) = 0 and so d I =. { } For part (): f u λg I{ } = wth ( ) ( ) wt u =, thn 0 λ F. Howvr, by usng quaton (), u = λg 0 and so u = λg I whch { } mpls ( { } ) d I >. xt, w chck whthr I consst of codwords of { } wght. Assum u = λ g + λ g I such that wt { } ( u ) =, thn w hav thr g, g H, g, g sh or g H and g sh. For ach of ths possblts, by usng quaton (), w wll obtan a st of quatons n trms λ and λ, and upon solvng wll gv th soluton λ = λ = 0 whch s of mpossbl. Thus, ( { } ) u = H H H H 0 = = Q.E.D. d I >. Fnally, consdr u = H, thn and so u H I{ } = and hnc ( { } ) d I =. From Lmma. and Rmark., w s that I { s a } (,5,) group cod for =, whch attan th snglton bound and so ar I s a (,,) group cod whch s not an MDS MDS cods. Howvr, { } cod. Thorm.. Lt, and b thos dmpotnts n FD, thn: d I = () ( {, } ) () ( {, } ) d I = for =,. Proof. By Lmma.7 and Lmma., w notc that ( ) d I for all {, }, =, and =,. For part (), f u = λ s + λ rs + λ 0 and λ 0, 5 5 thn by usng quaton () and (), u = ( λ + λ ) and u = ( λ λ ) 5 5 Malaysan Journal of Mathmatcal Scncs 07

8 Dns C. K. Wong & Ang M. H thn u = u = 0 f and only f λ = λ. Hnc, u = λ s + λ rs I. 5 5 {, } Thrfor, ( {, } ) d I =. For part (), th proof wll b smlar. W nd to fnd an lmnt wth wght qual to. It can b chckd that thr ar no codwords wth wght lss than n I. {, } W now chck that u = λ + λ r + λ r + λ s + λ rs + λ r s s a word of I. 5 {, } By usng quaton () and (), w obtan u ( λ λ λ λ λ λ ) = + + and 5 u = ( λ λ λ ) + ( λ + λ λ ) r + ( λ λ + λ ) r + λ λ λ s + λ + λ λ rs + λ λ + λ r s. ( 5 ) ( 5 ) ( 5 ) ] Thus, u = u = 0 f and only f ( λ λ λ λ λ λ ) + + = () 5 0 λ λ λ = 0 λ + λ λ = 0 λ λ + λ = 0 λ λ λ = 0 5 λ + λ λ = 0 5 λ λ + λ = 0 5 Th unqu soluton for () s λ = λ = λ 0 and for () s λ = λ = λ 0. Hnc, from (), λ + λ + λ λ λ λ = 0 whch mpls 5 that λ = λ 0. Thrfor, w obtan a nonzro soluton and so ( {, } ) d I =. Q.E.D. {, } () () In Thorm., w hav constructd two famls of group cods,,, MDS group cod and,, group cod for {, } I s a ( ) =,.. Cods dfnd ovr FD Lt H r r = = and so I s a ( ) D = H sh. ot that 0 Malaysan Journal of Mathmatcal Scncs K r r H = =. From Lmma., w s that D conssts of 5 rrducbl charactrs, n

9 Group Cods Dfn Ovr Dhdral Groups of Small Ordr whch cas, four of thm ar lnar charactrs and on s nonlnar charactr. Each of ths charactr wll corrspond to a unqu dmpotnt as follows: (a) Idmpotnts corrspond to lnar charactrs: χ = ( H + sh ), χ = ( H sh ), χ = ( r)( + s) K, and χ = ( r)( s) K. (b) Idmpotnts corrspond to th nonlnar charactr χ of dgr : 5 χ = 5 5 ( r ) = ( r ). Lt u = λ + λ r + λ r + λ r + λ s + λ rs + λ r s + λ r s b any 5 7 lmnt n FD such that λ F for =,,,, 5,, 7 and, thn u = λ () u u u ( ) = ( λ λ ) = = = 5 ( λ λ ) =,,5,7 =,,, ( λ λ ) =,,, =,,5,7 = = u = + r + + r + + r ( λ λ ) ( λ λ ) ( λ λ ) ( λ λ ) ( λ λ 5 7 ) s ( λ λ ) rs ( λ λ 5 7 ) r s ( λ λ ) r s] Lmma.. Lt {,,, } =, whr dmpotnts n FD f β, thn d ( I β ) =. (5) () (7) (),, and ar th lnar Proof. If β, thn thr ar four cass to b consdrd, whch ar β =,,, or. From Lmma.7, w only nd to show that d ( I β ) = for β =. If β =, thn,,, ar all n β. If u = λ g, λ 0 and wt ( u ) =, thn λ ( λ χ ( )) ( λ χ ( )) and u = ( λ χ ( g )) 0. Hnc, u λ g I β u = 0, u = g 0, u = g 0 d I β. = ndcats that ( ) Malaysan Journal of Mathmatcal Scncs 09

10 Dns C. K. Wong & Ang M. H xt, consdr u = λ + λ r, by usng quaton () to (7): = ( λ + λ ), = ( λ + λ ), = ( λ + λ ) and u ( λ λ ) u u u 0 λ λ = +. u = u = u = u = f and only f = 0. Clarly, u I β and so d ( I β ) =. Q.E.D. a (, β,) From ths lmma, w mmdatly conclud that β, thn I β s group cod. Furthrmor, I β s a MDS cod f and only f β =. Th nxt rsult can b provd by usng smlar mthod as Lmma.. Lmma.. Lt = { 5 } whr s th nonlnar dmpotnt n FD, thn 5 ( { } ) ( ) d I 5 = and { 5} dm I =. Furthrmor, lt u = λ g + λ g, λ 0 and λ 0 wth g g D, thn u I f and only f g, g { } Thorm.5. Lt {, } 5 5 H. = whr s any on of th lnar dmpotnts and s th nonlnar dmpotnt n FD, thn d 5 ( I ) = and so I s a (,, ) group cod. Proof. Wthout loss of gnralty, w only prov for th cas {, } =. By Lmma.7 and Lmma., w know that d ( I ). By th scond statmnt n Lmma., f u = λ g + λ g, λ 0 and λ 0, thn thr g, g n H or g, g n sh or on n H and th othr n sh wll not produc a codword n I. Ths follows from quatons () and () n whch always gvs th soluton λ = λ = 0. xt, for u = λ g + λ g + λ g, λ 0 and k k λ 0 and λ 0, w hav thr g, g, g all ls n H (rsp. sh ) or k k k g, g ls n H (rsp. sh ) but g ls n sh (rsp. H ). For both cass, by usng quatons () and (), u s not contand n I. Fnally, f u = + r + r + r, 0, 0, 0 and λ 0, thn u λ λ λ λ λ λ λ = ( λ + λ + λ + λ ) and ( λ λ ) ( λ λ ) u = + r Malaysan Journal of Mathmatcal Scncs

11 Group Cods Dfn Ovr Dhdral Groups of Small Ordr Thus, u = u = 0 f and only f λ + λ + λ + λ = 0 and λ λ = 0 and 5 λ λ = 0. Th only soluton for th abov s λ = λ 0 and λ = λ 0, So, λ + λ + λ + λ = 0 mpls that λ + λ = 0 and so λ = λ 0. Thus, w obtan a st of nonzro soluton and so u u I. In othr word, d ( I ) =. Q.E.D. Thorm.. Lt,,,, b th dmpotnts n FD, thn 5 d ( I {,, 5 } ) = and ( ) {,, 5 } dm I =, whr, =,,,,. Proof. By Lmma.7 and Thorm.5, w only nd to show that thr xsts a codword of wght n I, whr, =,,,,. Snc {,, } 5 most calculatons ar routnd, thn w only stat a codword of wght n ach group cod. () u = λ + λ r + λ r + λ r I. {,, } 5 () u = λ + λ r + λ s + λ r s I. 5 7 {,, } 5 () u = λ + λ r + λ rs + λ r s I. {,, } 5 (v) u = λ + λ r + λ rs + λ r s I. {,, } 5 (v) u = λ + λ r + λ rs + λ r s I. 7 {,, } 5 (v) u = λ + λ r + λ r + λ r I. {,, 5} Q.E.D. Corollary.7. d ( I ) {,,, 5 = and k ( ) } {,, k, } { },, k,,,, k. d I =, whr 5 Proof. Th proof s smlar to Thorm., and so wthout loss of gnalty,,, =,,, w consdr only = { }, n th cas of { } ( {,, } ) d I =, thus w may assum that th cod gnratd by {,,, } = has mnmum dstanc gratr than or qual to. 5 By usng quatons (), (5), () and (), t can b shown that no codword of wght, 5,, and 7 n I and so w only xhbt thr s an lmnt of {,, } wght n I. {,, } 5 5 Malaysan Journal of Mathmatcal Scncs

12 If =,,,...,, Dns C. K. Wong & Ang M. H u = λ + λ r + λ r + λ r + λ s + λ rs + λ r s + λ r λ for, thn u u u ( λ λ λ λ λ λ λ λ ) = , 5 7 ( λ λ λ λ λ λ λ λ ) = + + +, 5 7 ( λ λ λ λ λ λ λ λ ) = and 5 7 u r r r = 5 ( ) ( ) ( ) ( ) λ λ + λ λ + λ + λ + λ + λ ( λ λ 5 7 ) s ( λ λ ) rs ( λ λ 5 7 ) r s ( λ λ ) r s] Thus, u = u = u = u = 0 f and only f 5 λ + λ + λ + λ + λ + λ + λ + λ = 0, 5 7 λ + λ + λ + λ λ λ λ λ = 0, 5 7 λ λ + λ λ + λ λ + λ λ = 0, 5 7 λ = λ, λ = λ, λ = λ and λ = λ. 5 7 Hnc, () λ + λ + λ + λ = 0 5 () λ + λ λ λ = 0 5 () λ λ + λ λ = 0 5 Upon solvng () to (), w wll obtan nonzro soluton. Hnc, u I d I =. {,,, } and so ( {,,, } ) 5 5. Cods dfnd ovr FD 0 5 Lt H = r r = and so D = H sh. From Lmma., w s that D 0 0 conssts of rrducbl charactrs, n whch cas, two of thm ar lnar charactrs and th othr two ar nonlnar charactr. Each of ths charactr wll corrspond to a unqu dmpotnt as follows Malaysan Journal of Mathmatcal Scncs

13 Group Cods Dfn Ovr Dhdral Groups of Small Ordr (a) Idmpotnts corrspond to lnar charactrs: ( H sh ) χ = H sh 0 0 χ = + and ( ) (b) Idmpotnts corrspond to th nonlnar charactr χ of dgr : 5 χ = ( ( )( ) ( )( )) r + r + 5 r + r and 0 χ = ( ( )( ) ( )( )) r + r + 5 r + r 0 W sumarz our rsults n th followng thorm. Indd most of thm can b provd by usng smlar argumnt as for group cods n FD and FD. Thorm.. Lt = {, } and {, } whch s dfnd as abov. L = b all dmpotnts n FD 0 () If β such that β =, thn d L ( I β ) = and dm( I β ) = 9. () If β such that β =, thn d ( I β ) = and ( ) dm I β =. Furthrmor, f v I β wth wt( v ) = thn supp( v) D ' or 0 supp( v) sd '. 0 () If β = {, } for =, and =,, thn d ( I β ) = and dm( I β ) = 5. Furthrmor, f v I β wth wt( v ) = thn (v) If supp( v) D ' or supp( v) sd 0 0 '. β =, thn ( ) 0 d I β = and ( ) v wth wt( v ) = 0 thn supp( v)= D ' sd '. 0 0 I β (v) If {,, } β = for, dm I β =. Furthrmor, f =, thn d ( I β ) = and ( ) dm I β =. Furthrmor, f v I β wth wt( v ) = thn supp( v) D ' or 0 supp( v) sd '. (v) If {,, } 0 β = for, =, thn d ( I β ) = 0 and ( ) dm I β =. Furthrmor, f v wth wt( v ) = 0 thn supp( v)= D ' sd '. 0 0 I β Malaysan Journal of Mathmatcal Scncs

14 Dns C. K. Wong & Ang M. H. Cods dfnd ovr FD D conssts of sx rrducbl charactrs, four ar lnar charactrs and two ar nonlnar charactrs. Each of ths charactr wll corrspond to a dstnct dmpotnt n th followng way. (a) Idmpotnts corrspond to lnar charactrs: 5 5 χ = ( r ( + s) ), χ = ( r ( s) ), = 0 = χ ( ( r) ( s) ) χ = ( ) ( ) r s = 0 = 0 = + and ( ) (b) Idmpotnts corrspond to nonlnar charactrs: χ = 5 5 ( r r ) ( + r ) and χ = ( r r )( r ) Lt 7 u = λ r + λ r s b any lmnts n FD such that = = 7 λ F, thn u u u u = ( λ ) = ( λ ( λ )) = = 7 λ ( λ ) = + ( ) =,,5,7,9, =,,,,0, ( λ ( λ )) =,,5,,0, =,,,7,9, = + = + u = = ( λ λ λ λ λ λ ) ( λ λ λ λ λ λ ) r ( λ λ λ λ λ λ 5 ) r ( λ λ λ λ λ λ 5 ) r 5 ( λ λ λ λ λ λ 5 ) r ( λ λ λ λ λ λ 5 ) r ( λ λ λ λ λ λ ) s ( λ λ λ λ λ λ ) r s ( λ λ λ λ λ + λ 7 ) r s + ( λ λ + λ λ λ + λ ) r 5 s ] u = ( λ λ λ λ λ λ ) ( λ λ λ λ λ λ ) r ( λ λ λ λ λ λ 5 ) r ( λ λ λ λ λ λ 5 ) r 5 ( λ λ λ λ λ λ 5 ) r ( λ λ λ λ λ λ 5 ) r ( λ λ λ λ λ λ ) s ( λ λ λ λ λ λ ) r s ( λ λ λ λ λ λ 7 9 ) rs 0 ( λ + λ + λ λ λ λ ) r s ( λ λ λ λ λ λ ) r s ( λ λ λ λ λ λ ) r 5 s ] (9) (0) () () () () Malaysan Journal of Mathmatcal Scncs

15 Group Cods Dfn Ovr Dhdral Groups of Small Ordr Thorm.9. Lt = {,,, } and {, } L FD whch s dfnd as abov. () If β, thn d L ( I β ) = and ( I β ) = b all dmpotnts n 5 dm = β. () If β and β =, thn d ( I β ) = and dm( I β ) =. () If β = {, 5 }, thn d ( I {, } ) = only f = or, and ( ) 5 {, } only f = or. Furthrmor, dm( I β ) = 7. (v) If β = {, }, thn d ( I {, } ) = only f = or, and ( ) {, } only f = or. Furthrmor, dm( I β ) = 7. (v) d ( I ) = and dm( I β ) =. (v) d ( I{,, } ) = d ( I{,, } ) = and dm 5 ( I ) β =. (v) d ( I{,, } ) = d ( I{,, } ) = and dm 5 ( I ) β =. (v) d ( I ) = d ( I {,, } {,, } ) = d ( I{,, } ) = d ( I{,, } ) = and ( ) (x) d ( I {,, } ) = for =,,,, and dm 5 ( I ) β =. (x) d ( I {,,, 5 } ) =, whr,, k =,,,, k, and dm( I ) 7 k β =. (x) d ( I{, } ) = d ( I{, } ) = and dm ( I β ) =. (x) d ( I{, } ) = d ( I{, } ) = and dm ( I β ) =. (x) If β and β =, thn d ( I β ) = and dm( I ) β =. L d I = 5 d I = ) dm I β =. REFERECES Brman, S. D. 97. Smsmpl Cyclc and Ablan Cods, II. Kbrntka. : -0. Brman, S. D. 99. Paramtr of Ablan Cods n th Group Algbra KG of G = <a> <b>, a p = b p =, p s prm, ovr a fnt fld K wth a prmtv p th root of unty and rlatd MDS-Cods. Contmpary Math. 9. Malaysan Journal of Mathmatcal Scncs 5

16 Dns C. K. Wong & Ang M. H How Guan Aun and Dns Wong Ch Kong. 00. Group Cods Dfnd Usng Extra-Spcal p-group of Ordr p. Bull. Malays. Math. Sc. Soc. : 7: Hurly, P. and Hurly, T Modul cods n group rngs. Proc. IEEE Int. Symp. on Informaton Thory (ISIT). Hurly, P. and Hurly, T Cods from Zro-dvsors and unts n group rngs. Int. J. Inform, and Codng Thory. : Isaacs, I. M. 99. Algbra, A Graduat Cours. Calforna: Brooks/Col Publshng. Pacfc Grov. Jams, G. D. and Lbck, M. W. 99. Rprsntatons and Charactrs of groups. Cambrdg Unvrsty Prss. Macwllam, F. J. 99. Cods and dals n group algbras, n Combnatoral Mathmatcs and ts Applcatons, R.C. Bos and T.A. Dowlng, ds., Chapl Hll: Unv. orth Carolna Prss, 7-. Passman, D. S Th Algbrac Structur of Group Rngs. w York: Wly. Sloan,. J. A. and Macwllam, F. J. 97. Th Thory of Error Corrctng Cods. Amstrdam, thrlands: orth-holland. Malaysan Journal of Mathmatcal Scncs