Fakultas Matematika dan Ilmu Pengetahuan Alam, Institut Teknologi Bandung, Bandung, 40132, Indonesia *

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1 MacWllams Equvalece Theoem fo he Lee Wegh ove Z 4 leams Baa * Fakulas Maemaka da Ilmu Pegeahua lam, Isu Tekolog Badug, Badug, 403, Idoesa * oesodg uho: baa@mahbacd BSTRT Fo codes ove felds, he MacWllams equvalece heoem gve us a comlee chaacezao whe wo codes ae equvale osdeg he moa ole of he Lee wegh codg heoy, oe would lke o have a smla esuls fo codes ove ege esdue gs equed wh he Lee wegh We would lke o ove ha he lea somohsms bewee wo codes Z m ha s esevg he Lee wegh ae exacly he mas of he fom f ( x, x,, xm) ( u x (), u x (),, umx ( m) ) whee u,, {,} u m ad S The oblem s sll lagely oe Wood oved he esul fo codes ove Z whee s a owe of o 3 I hs ae we ove he esul fo me of he fom 4 whee s me (Keywods: MacWllams equvalece, exeso heoem, Lee wegh, codes ove gs) INTRODUTION Le ad be lea codes ove fe felds F The wo codes ae equvale f hee s a lea somey bewee ad, ha s, f hee s a lea ma f : ha eseves he Hammg wegh of evey codewod x The MacWllams equvalece heoem [] saes ha lea somees ae mas of he fom whee u 0 ad S f ( x,, x ) ( u x,, u x ), m () m ( m) The celebaed esuls of Hammos, aldebak, Kuma, Sloae, ad Sole [] show ha a class of omal olea codes called he Kedock ad Peaaa ca be cosdeed as lea codes ove Z 4 equed wh he Lee wegh Sce he, he Lee wegh had become a moa wegh codg heoy besdes he Hammg wegh Geealzg Macwllams Theoem fo codes ove gs equed wh he Hammg wegh oved o be successful Wad ad Wood [3] gave a ew oof of he MacWllams Theoem fo codes ove felds usg a chaace heoy echque Gefeah ad Schmd [4] oved he same esul usg a combaoal mehod Lae, usg he same chaace heoy echque, Wood [5] oved ha he MacWllams Theoem holds fo a class of gs called he Fobeus gs I 008, Wood [6] oved ha hee was o lage class of gs whch he heoem holds, by showg ha o have he MacWllams heoem ove a g, he g s ecessay o be Fobeus ohe deco of geealzg Macwllams heoem s by cosdeg weghs ohe ha he Hammg wegh Goldbeg [7] oved he heoem fo he symmezed wegh comoso Wood [8] gave a suffce codo whe he heoem holds fo geeal weghs ems of vebly of some max O he same ae, Wood showed ha usg hs ceo, he MacWllams heoem fo he Lee wegh Z, fo s a owe of o 3 o me of he fom whee s also me I hs ae, we wll show ha fo me, he max he ceo of Wood above, has a ccula sucue By usg hs sucue, we ove ha he Macwllams heoem fo he Lee wegh holds fo all mes of he fom 4 whee s also me DEFINITIONS lea code ove a commuave g Z of legh m s a Z -submodule of Z m Fo k Z he Lee wegh of k, deoed by Lk ( ) s defed as L( k) : m{ k, k} Fo examle fo 3, 4,5 Z 7 we have L(3) L(4) 3 ad L(5) esecvely We ca exed hs defo

2 of he Lee wegh o ay veco x ( x,, x m ) Z m by defg m L( x) : L( x ) Le, be codes ove Z lea ma f : s called a Lee somey f fo evey xwe have L( x) L( f ( x)) We say ha he equvalece heoem fo he Lee wegh holds fo Z f fo evey Lee somey f : hee s a emuao S ad u,, {,} u such ha f ( x, x,, x ) ( u x,, u x ) () ( ) osde Z as{0,,, } Fo k,, /, defe he Lee class of k by : { k, k} { k, k} Noe ha k k s he se of all elemes Z of he Lee wegh k, ha s L( x) k f ad oly f x k Noce ha f x, y k fo some k he L( xa) L( ya) fo evey a Z RITERION OF WOOD Fo evey g Z assocae a / / max ( a ) defed by a : L( ab) whee a, b, whee he mullcao a b s a mullcao Z I s easy o see ha he defo of s deede of he choce of a ad b Fo examle, he max assocaed o he g Z 7 s L() L( ) L( 3) 3 L() L() L(3) 3 L(3) L(3 ) L(33) 3 The esco of he esul of Wood [8] o he Lee wegh ad o he g Z gves a suffce codo fo Z o have a equvalece heoem fo he Lee wegh The ceo s gve ems of he max assocaed o Z Teoema Le be he max assocaed o Z If s veble (as max ove eals) he he equvalece heoem holds fo Z Poof : See [8] (Pooso ) Fo examle, he max assocaed o he g Z 7 s veble ad hece he equvalece heoem holds fo Z 7 THE STRUTURE OF MTRIX Noce ha he vebly of s vaa ude he ow ad colum emuaos We wll show ha whe s me,afe seveal ow ad colum emuaos, he max has a ccula sucue Le 3 be me ad be a mve oo modulo, we / Oe ca oba he max wo ses Fs, we ca make a mullcao able ove Z whee he colums ad ows ae dexed by,,, The he ees of he max ae he Lee wegh of he e ees he mullcao able Fo examle, fo Z 7 he max ca be obaed as follows If he ows ae sead dexed by,,3 ad he colums ae dexed by 3,,, he he mullcao able ad he max ae gve below Sce he ew able ca be obaed by ow ad colum emuaos of he evous able, he he ew max ca also be obaed by ow ad colum emuaos of he old max Sce we ae oly eesed he vebly of he max, we o loge have o dffeeae bewee he old ad he ew max ad smly call hem boh Pooso Le be me The he max s ccula Poof: Le be a mve oo modulo ad le : / Sce, fo evey k 0,,, we k k k k have {, } {, } s oe of he Lee class s fo 3

3 some s Hece,,,, ae eeseaves of,,, (o ecessaly ha ode) Now cosde he mullcao able whee he ows ae dexed by,,, ad he colums ae dexed by,,, 3 Usg he fac ha ad L( a) L( a) fo evey a Z, we have L L L L L L L L L L L L Theefoe s ccula ( ) ( ) () 3 L L L L L L L L L L L L ( ) ( ) () () ( ) ( ) ( ) () ( ) 3 Gve a ccula max of sze whee he fs ow of s ( a0,, a ) defe a olyomal P, called he esee of by 0 P ( x) a x a x a The followg heoem hels us defyg he vebly of a ccula max Pooso 4 (Ka [9], oollay 0) Le be a cucula max wh a esee olyomal P The followg ae equvale s veble P ( ò ) 0 fo all comlex oo ò of x 3 P ( x ) ad x ae elavely me I he case whee he sze of he ccula max s me umbe, we have he followg esul Pooso 4 (Ka [9], Pooso 3) Le be a ccula max of sze whee s a me umbe Le c0 c c 0 (,,, ) Z be he fs ow of If c 0 ad c 0, c,, c ae o all he same, he s veble By usg he above ooso, we have a ew oof of he esul dscoveed by Wood [8] oollay 5 If whee s a me umbe, he he equvalece heoem holds fo Z Poof: Le be he ccula max assocaed wh Z Ths max has sze ad he ees of s fs ow s a emuao of {,,, } By Pooso 4 we have ha s veble ad hece by he ceo of Wood he esul follows MIN RESULT I hs seco we wll cosde he equvalece heoem fo Z 4 whee s a odd me umbe I s kow ha whe ad 4 ae me, he s a mve oo modulo 4 (see [0] age 497 fo examle) I follows ha he fs ow of he max assocaed o Z 4 s (, L(), L( ),, L( )) Le P be he esee olyomal of We wll show ha s veble by showg ha P ( x ) ad x has o commo oo Theoem 6 The equvalece heoem holds fo Z 4 whee ad 4 ae me Poof: The comlex oos of x fom a cyclc gou of ode If s a oo of x, he he ode of s oe of,, o By he evous obsevao, we kow ha P ( x) x L() x L( ) s he esee max of We wll show ha hee s o oo of x whch s also a oo of P ( x ) We cosde seveal cases: 4

4 od( ) I hs case we have Sce he se {, L(),, L( )} s equal o {,,, }, he ( )( ) P () 0 od( ) The ad P ( ) L( )( ) 0 L L L L If P ( ) 0 he I follows ha () () ( ) ( ) odd eve L( ) L( ) ( ) L( ) L( ) odd Bu hs s mossble sce he lef had sde s odd whle he gh had sde s eve od( ) The fo all If P( ) 0 he ( ) 0 Hece s a oo of L( ) L( ) 0 Q( x) a x whee 0 a L( ) L( ) Sce he mmal olyomal ove Z of s m( x) x 0, he m( x) Q( x ) Bu mx ( ) ad Qx ( ) ae of he same degee Hece Q( x) K m( x) fo some osve ege K I acula, fo all we have L( ) L( ) K Now K L( ) L( ) 0 ( ) Hece L( ) K ad we coclude ha L( ) Noe ha fo all m Z we have L( m) m mod I follows ha L( ) mod 4 ( ) 4 mod 4 Squag boh sdes we have mod 4 Sce s a geeao of Z 4, he 4 whch s mossble sce 4 fo odd me od( ) The If P ( ) 0 0, he ( ) 0 L( ) L( ) Sce he mmal olyomal of hs case s m( x), he fo all we have L( ) L( ) L( ) L( ) I acula fo 0 we have L( ) L( ) L() L() 3 Reducg he equao modulo 4 we have 3 mod 4 Squag boh sdes, mod 4 So ehe 9 mod mod 4 o Ths mles ha 4 0 o 4 8 whch s mossble fo 3 I ay case we have ( ) 0 ad hece s veble Theefoe Z 4 sasfes he equvalece heoem accodg o he ceo of Wood 5

5 ONLUSION ND ONJETURE By usg Wood's ceo, we ae able o show he equvalece heoem holds fo he Lee wegh fo codes ove Z 4 whee ad 4 ae me By usg he ccula sucue of he max, he equvalece heoem holds ue fo Z f ad oly f P ( x ) ad x has o cosa commo dvso Z[ x ] comue seach o MPLE shows ha fo he fs 000 me umbes he olyomal P ( x ) s educble ove Z ad hece fo hose mes, he g Z sasfes he equvalece heoem Based o hs obsevao we sogly beleve ha he equvalece heoem holds fo Z ad fo ay me KNOWLEDGEMENT Ths eseach s suoed by Rse KK ITB 05 REFERENE MacWllams, FJ (96) ombaoal Poblems of Elemeay bela Gous, Docoal dsseao, Radclffe ollege Roge Hammos J, P Vay Kuma, Robe aldebak, Nel J Sloae, ad Pack Solé The z 4-leay of kedock, eaaa, goehals, ad elaed codes Ifomao Theoy, IEEE Tasacos o, 40():30 39, Wad, H N, & Wood, J (996) haaces ad he equvalece of codes Joual of ombaoal Theoy, Sees, 73(), Gefeah, Macus ad Schmd, Sefa E Fe-g combaocs ad MacWllams equvalece heoem I Joual of ombaoal Theoy, Sees, ages 7 8 Elseve, Wood, J (999) Dualy fo modules ove fe gs ad alcaos o codg heoy meca oual of Mahemacs, (3), Wood, J (008) ode equvalece chaacezes fe Fobeus gs Poceedgs of he meca Mahemacal Socey, 36(), Goldbeg, D Y (980) geealzed wegh fo lea codes ad a W-MacWllams heoem Joual of ombaoal Theoy, Sees, 9(3), Wood, J Exeso heoems fo lea codes ove fe gs I led algeba, algebac algohms ad eo-coecg codes, ages Sge, Iw Ka ad Saago R Smaca O ccula maces Noces of he MS, 59(3): , 0 0 Thomas Koshy Elemeay umbe heoy wh alcaos cademc ess, 00 6