Perfect Constant-Weight Codes

Transcription

1 56 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER 004 Prfct Costat-Wght Cods Tuv Etzo, Fllo, IEEE, ad Mosh Schartz, Mmbr, IEEE Abstract I hs porg or from 973, Dlsart cocturd that thr ar o otrval prfct cods th Johso schm. May attmpts r mad, durg th yars hch follod, to prov Dlsart s coctur, but oly partal rsults hav b obtad. W survy all ths attmpts, ad prov som rsults havg th sam flavor. W also prst a mthod, tag a dffrt approach, hch hop ca lad to th sttlg of ths coctur. W sho ho ths mthod ruls out sts of paramtrs as ll as spcfc gv paramtrs. Idx Trms Costat-ght cods, Johso schm, -rgular cods, prfct cods, Str systms. I. INTRODUCTION I a gv mtrc, cods hch atta th sphr-pacg boud th mtrc ar calld prfct. Such cods hav alays dra th attto of codg thorsts ad mathmatcas. I th Hammg schm, all prfct cods ovr ft flds ar o []. Thy xst for oly a rlatvly small umbr of paramtrs, hl for othr paramtrs t as provd [] [4] that thy caot xst. Th oxstc proof s basd of Lloyd s polyomals. For o-fld alphabts oly trval cods ar o ad by smlar mthods t as provd [5] that for most othr paramtrs thy doot xst. Costat-ght cods ar buldg blocs for gral cods th Hammg schm. Thy ar also of trst a d rag of aras [6] [0]. A atural qusto s hthr thr xst prfct costatght cods. I th 970s ad 980s, most or o costat-ght cods cosdrd oly th bary cas. A bary costat-ght cod has thr paramtrs lgth, costat ght, ad mmum Hammg dstac d (Hammg dstac ll b calld H-dstac for short). If df th dstac bt to ords x ad y of ght, as half thr H-dstac, obta a mtrc hch s calld th Johso mtrc, ad th dstac s calld th J-dstac. It s vry covt to dscrb th Johso schm trms of sts. Wth th Johso schm assocat th Johso graph J(; ). Th vrtx st V of th Johso graph cossts of all -substs of a fxd -st. Tosuch -substs ar adact f ad oly f thr trscto has sz 0. A cod C of such -substs s calld a -prfct cod J(; ) (or th Johso schm) f th -sphrs th ctrs at th codords of C form a partto of V. I othr ords, C s a -prfct cod f for ach lmt v V thr xsts a uqu lmt c C such that th J-dstac bt v ad c s at most. Thr ar som trval prfct cods J(; ). ) V s 0-prfct. ) Ay fvg, v V,s-prfct. 3) If, odd, ay par of dsot -substs s -prfct th ( 0 ). It as cocturd by Dlsart [], that ths ar th oly prfct cods J(; ). Mauscrpt rcvd May, 003; rvsd May 3, 004. Th matral ths corrspodc as prstd part at th 99st AMS Mtg, Uvrsty of North Carola, Chapl Hll, NC, Octobr 003. Th authors ar th th Computr Scc Dpartmt, Tcho Isral Isttut of Tchology, 3000 Hafa, Isral (-mal tzo@cs.tcho.ac.l; moosh@cs.tcho.ac.l). Commucatd by C. Carlt, Assocat Edtor for Codg Thory. Dgtal Obct Idtfr 0.09/TIT Th ma purpos of ths corrspodc s to prst a tchqu hch could lad to th sttlg of th xstc qusto of prfct cods th Johso schm. I Scto II, gv a short survy of th o rsults ad tchqus cocrg th xstc of prfct cods th Johso schm. Paramtrs for hch prfct cods caot xst ar summarzd. I Scto III, prst som rsults usg smlar tchqus, hch rul out mor paramtrs for hch prfct cods caot xst. I Scto IV, prst a tchqu toprov that -prfct cods do ot xst J(; ). W sho hch gral paramtrs ca b ruld out by th tchqu. I Scto V, summarz th paramtrs for hch thr ar o -prfct cods J(; ). W alsodscrb a computr sarch, usg th tchqu, th hch r abl to sho th oxstc of -prfct cods J(; ) for ay gv spcfc,, ad that chcd. Cocluso s gv Scto VI. II. SURVEY OF KNOWN RESULTS I hs or from 973, Dlsart rot [, p. 55] Aftr havg rcalld that thr ar vry f prfct cods th Hammg schms, o must say that, for <<, thr s ot a sgl o o th Johso schms. It s tmptg to rs th coctur that such cods do ot xst. Crta rsults cotad th prst or could b usful to attac ths problm; spcally th gralzd Lloyd thorm of sc. 5.. ad thorm 4.7 about t-dsgs. Idd, Dlsart omttd th trval prfct cods ( ll omt thm too, so h say prfct cods ma otrval prfct cods) ad hs coctur o th oxstc of prfct cods th Johso schm has provdd lots of groud for rsarch th t yars hch follod. Du to th fact that th Hammg schm all paramtrs for hch prfct cods xst r o, spcal mphass as gv to th Johso schm. Hovr, most rsarch fald to produc sgfcat rsults. I th Hammg schm, th trval cods, th Hammg cods, ad th to Golay cods, ar th oly prfct cods ovr GF (q). Thr ar o prfct cods th othr paramtrs [] [4] (s also [] for th dtald proof). Morovr, for most paramtrs, t s o that thr ar o prfct cods ovr o-fld sz alphabts th Hammg schm [5]. Bggs [] shod that th atural sttg for th xstc problm of prfct cods s th class of dstac-trastv graphs. Lt 0 b a coctd graph. W dot by d 0(x; y) th lgth of th shortst path from x to y. 0 s sad tob dstac-trastv f, hvr x, x 0, y, y 0 ar vrtcs th d 0 (x; x 0 )d 0 (y; y 0 ), thr s a automorphsm of 0 th (x) y ad (x 0 )y 0. Bggs [] clams that th class of dstac-trastv graphs cluds all trstg schms, such as th Hammg schm ad th Johso schm. Ths graphs ar cotad aothr class of graphs. 0 s sad tob dstac-rgular f thr ar tgrs a, b, c (0 d, hr d s th damtr of 0) th th follog proprty hvr x ad x 0 ar vrtcs th d 0 (x; x 0 ), th y d 0 (x; y) ad d 0 (x 0 ;y) a; b; or c dpdg o hthr 0,,or +. A dstac-trastv graph s obvously dstac-rgular. Lt 0 b a dstac-rgular graph th a vrtx st V. A subst X of V s calld a atcod th damtr,f s th maxmum dstac occurrg bt vrtcs of X. Atcods th damtr havg maxmal sz ar calld optmal atcods. Th follog thorm s du to Dlsart [] /04$ IEEE

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER Thorm Lt X ad Y b substs of V such that th ozro dstacs occurrg bt vrtcs of X do ot occur bt vrtcs of Y. Th X Y V. Bggs [] dvlopd a gral thory ad a smpl crtro for th xstc of prfct cods a dstac-trastv graph. H shod that ths crtro mpls Lloyd s thorm, hch s usd th Hammg schm to prov th oxstc of prfct cods all cass. Baa [3] provd th oxstc of -prfct cods J(0;) ad J(+;) for. H usd a aalog to Lloyd s thorm ad som umbr-thortc rsults. Hammod [4] mprovd ths rsult by provg th follog. Thorm Thr ar oprfct cods J( 0 ;), J( 0 ;), J( +;), ad J( +;). Hovr, th most sgfcat rsult, th frst 0 yars follog Dlsart s coctur, as gv 983 by Roos [5]. Thorm 3 If a -prfct cod xsts J(; ), th + ( 0 ) Th proof of Roos as basd o th follog thory gv by Dlsart [] by usg Thorm, Roos otcd that f a -prfct cod xsts, th th -sphrs should b optmal atcods th damtr. H procdd to fd atcods J(; ) ad obtad hs rsult by comparg thm to th -sphrs. I Scto III, ll gv a dffrt smpl proof to Thorm 3. Thr s a spcal trst th tchqu of Roos ad Thorm of Dlsart as ll dscuss Scto II-A. It too mor tha 0 yars to obta rsults. Etzo [6] too a approach. H provd that f thr xsts a otrval -prfct cod C J(; ), th may Str systms ar mbddd C. A Str systm S(t; ; ) s a collcto of -substs (calld blocs) ta from a -st, such that ach t-subst of th -st s cotad xactly o bloc. Th follog thorms ar ll o (s [] for rfrc). Thorm 4 If thr xsts a Str systm S(t; ; ) for t, th thr xsts a Str systm S(t 0 ;0 ;0 ). Thorm 5 A cssary codto for a Str systm S(t; ; ) toxst, s that th umbrs 0 t 0 must b tgrs, for all 0 t. 0 t 0 Usg Etzo s approach, th cssary codtos of Thorm 5 mply cssary codtos for th xstc of prfct cods th Johso schm. Morovr, Etzo dvlopd a cocpt calld cofgurato dstrbuto, hch s a to th cocpt of ght dstrbuto for cods th Hammg schm. Usg ths cocpt, combd th th cssary codtos drvd from Str systms, may paramtrs r foud, for hch -prfct cods do ot xst J(; ). W summarz th ma rsults gv [6]. Lmma If C s a -prfct cod th Johso schm, th ts mmum H-dstac s 4 +. A (; d; ) cod s a cod of lgth, costat ght, ad mmum H-dstac d. A(; d; ) dots th maxmum sz of a (; d; ) cod. Th follog lmma s a trval obsrvato. Lmma If C s a -prfct cod J(; ), th A(; 4 +;)C Hcforth, lt N f; ;...;g b th -st. From a Str systm S(t; ; ) costruct a costat-ght cod o coordats as follos. From ach bloc B costruct a codord th s th postos of B ad 0 s th postos of NB. Ths costructo lads to th follog ll-o thorm (s rfrc [7]). Thorm 6 ( 0 ) ( 0 t +) A(; ( 0 t +);) ( 0 ) ( 0 t +) f ad oly f a Str systm S(t; ; ) xsts. From Thorm 6 ad Lmmas ad, mmdatly fr th follog rsult. Lmma 3 If C s a -prfct cod J(; ) hch s alsoa Str systm, th t s a Str systm S( 0 ; ; ). Th xt lmma s a smpl obsrvato of a cosdrabl us. Lmma 4 Th complmt of a -prfct cod J(; ) s a -prfct cod J(; 0 ). Fally, d a f mor dftos hch ll us th proofs of th oxstc thorms th squl. For a gv partto of N totosubsts A ad B, such that A ad B 0, lt cofgurato (; ) cosst of all vctors th ght th postos of A ad ght th postos of B. For a -prfct cod C J(; ), say that u C J-covrs v V f th J-dstac bt u ad v s at most. I th squl, ll us a mxd laguag of st otato ad vctor otato. It should b udrstood from th cotxt hch o ar usg, ad ho to traslat th to dffrt otatos. Th follog rsults r provd [6]. Thorm 7 If a -prfct cod xsts J(; ), th a Str systm S(+; +;) ad a Str systm S(+; +;0) xst. Corollary If a -prfct cod xsts J(; ), th a Str systm S(; +; 0 +)ad a Str systm S(; +; 0 0 +)xst. Corollary If a -prfct cod xsts J(; ), th 0 (mod +)ad hc +dvds 0. Thorm 8 Excpt for th Str systms S(;;) ad S(; ; ), thr ar omor Str systms hch ar alsoprfct cods th Johso schm. Thorm 9 Thr ar o -prfct cods J( + p; ), p prm, J( +p; ), p prm, p 6 3, ad J( +3p; ), p prm, p 6 ; 3; 5. If comb Lmma 4 th th fact that th J-dstac bt ords of a -prfct cod s at last +, gt th follog. Corollary 3 If a -prfct cod xsts J(; ), th + ad 0 +. To gv th radr th flavor of th mthods usd [6], us smlar mthods to provd a much smplr proof of Thorm 3, ad toprov that thr ar v mor Str systms mbddd prfct cods th Johso schm (ths rsults r also prstd [7]). Thorm 0 If a -prfct cod xsts J(; ) th + ( 0 )

3 58 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER 004 Proof Assum C s a -prfct cod J(; ). W partto N totosubsts A ad B, such that A 0 +, B 0, ad thr s a codord of cofgurato (+;00). Clarly, th J-dstac bt a vctor from cofgurato (+;00) ad a vctor from cofgurato (+0; 00+), 0 <, s strctly lss tha +,soc dos ot hav ay codord from cofgurato (+0; 0 0+). Thrfor, all th vctors from cofgurato (;0) ar J-covrd by codords from cofgurato ( +;0 0 ). To J-covr ach vctor from cofgurato (;0 ) xactly oc must hav xactly 0+ codords from cofgurato (+ + ; 0 0 ). Sc th mmum J-dstac of C s +,to codords from cofgurato ( +;0 0 ) caot trsct th zros of part B. Hc, 0 0+, hch s quvalt to + ( 0 ) +. Thorm If a -prfct cod xsts J(; ) ad < ( 0 )( +), th a S(;+;0 +)xsts. Proof Assum C s a -prfct cod J(; ). As th proof of Thorm, partto N totosubsts A ad B, such that A 0 +, B 0, ad thr ar 0+ + codords from cofgurato ( +;0 0 ). Sc <( 0 )( +),.., 0+ + < 0, hav at last o coordat B hch has os all th codords from cofgurato ( +; 0 0 ). W rmov ths coordat from B toobta B ad o t to A toobta A. A 0 +, B 0, ad C dos ot hav ay codord from cofgurato ( +0 ; 0 0 +), > 0. Thrfor, all th vctors from cofgurato (; 0 ) ar J-covrd by codords from cofgurato ( +;0 0 ). Sc ach vctor from cofgurato (;0) must b J-covrd by xactly o codord from cofgurato ( +;0 0 ), t follos that part A of th codords from cofgurato ( +;0 0 ) forms a Str systm S(;+;0 +). Corollary 4 If a -prfct cod xsts J( + a; ), a 0, th a S(;+;+)xsts. Mart also xamd th xstc problm h h cosdrd compltly rgular substs hs Ph.D. dssrtato [8]. H foud that f, th prfct cods must oby som umrcal formulas. Etzo [7] polshd som of th rsults from [6]. Rctly, Shmabuuro [9] shod that, as a applcato of Etzo s rsults, o ca obta that thr ar o prfct cods J( + 5p; ), p prm, p 6 3; J( + p ;), p prm. A. Str Systms ad Prfct Cods Str systms play a mportat rol rulg out th xstc of -prfct cods J(; ). Morovr, th Str systms S(;;), odd, ad S(; ; ), ar amog th trval prfct cods th Johso schm. Thorm 8 stats that thr ar o mor Str systms hch ar also prfct cods th Johso schm. By Thorm 6, ay Str systm s a optmal costat-ght cod. Obvously, ay -prfct cod J(; ) s alsoa optmal costat-ght cod. Sc a Str systm S(t; ; ) ach t-subst appars xactly o bloc, t ould b atural to th that Str systms ar prfct cods of som d. By Lmma, a -prfct cod J(; ) s a optmal (; 4 + ;) cod ad th sz of such a cod s clarly (; ; ) 0 0 By Thorm 6, a Str systm S(0; ; ) s a optmal (; 4+ ;) cod ad ts sz s s(; ; )!()! ( 0 +)!! If (; ; ) >s(; ; ) th thr s o -prfct cod J(; ) ad f s(; ; ) >(; ; ) th thr s ostr systm S( 0 ; ; ). Not that (; ; ) s(; ; ) h 4 + hch cas a trval Str systm S(; +; 4 +)ad a trval -prfct cod J(4 +; +)xst. No bouds for thr -prfct cods J(; ) or Str systms S( 0 ; ; ) ca b drvd from ths codtos. Rctly, Ahlsd, Ayda, ad Khachatra [0] gav a trstg dfto of damtr-prfct cods (D-prfct cods). Thy xamd a varat of Thorm. Lt 0 b a dstac-rgular graph th a vrtx st V. IfA s a atcod 0, dot by D(A) th damtr of A. No lt A 3 (D) maxfa D(A) Dg Thorm If C s a cod 0 th mmum dstac D +, th C V A 3 (D) 0. Thy cotud th th follog dfto for prfct cods. A cod C th mmum dstac D +s calld D-prfct f Thorm holds th qualty. Ths s a gralzato of th usual dfto of -prfct cods as -sphrs ar atcods th damtr. Ths dfto for prfct cods troducd som classs of prfct cods. Th trstg classs ar thos of cods hch atta som classcal boud. I th Johso schm, t as provd that all Str systms ar D-prfct, thus shog mor coctos bt Str systms ad prfct cods. III. NEW RESULTS I ths scto, cotu to prov rsults o th structur of -prfct cods J(; ). As a rsult, dtfy mor paramtrs of,, ad, hch such cods caot xst. A. N Uppr Boud o W frst sho that o otrval -prfct cod achvs Roos boud th qualty. Ths smgly slght mprovmt has may applcatos. Thorm 3 If thr xsts a -prfct cod J(; ) th + <(0) Proof If <th by Corollary 3 th clam s obvous. Assum C s a -prfct cod J(; ), hr + a, a 0, ad (0) +.Ifa 0, th ad (0) + mply that +,.., C s a trval prfct cod. By Thorm, hav that thr ar oprfct cods J(+;), ad, thrfor, a. Lt b +; by Corollary, a 0 0(modb), ad hc, b a. W substtut b +ad + a, ( 0 ) +, ad obta ab 0 a +b0. By prvous thorms, th follog Str systms must xst. By Corollary, thr xsts a Str systm S(;b+;ab0a + b +). Thus, by Thorm 5, ( ) must b a tgr. ( ) By Corollary thr also xsts a Str systm S(;b +; ab + b +). Thus, by Thorm 5, ( ) must b a tgr. ( )

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER By Corollary 4, thr also xsts a Str systm S(;b +; ab 0 a +b +). Thus, by Thorm 5, ( ) must b a ( ) tgr. Thrfor, ab+b+ b+ 0 s a tgr, ad hc, ab0a+b+ b+ a 0 a b +a + ab b + a 0 a b +a + ab 0 (mod b +) But, sc b 0 (mod b +), hav that W alsohav that ab0a+b+ b+ 3a + a 0 (mod b +) () 0 s a tgr, ad hc, ab0a+b+ b+ ab 0 a +3b + b + ab 0 a +3b + 0 (mod b +) Aga, b 0 (mod b +)hch mpls that By () ad () hav that 4a + 0 (mod b +) () 8(3a + a) 0 (6a 0 )(4a +) 0 (mod b +) (3) But, 8(3a +a)0(6a0)(4a+) ad clarly s ot dvsbl by b +, a cotradcto. Hc, <( 0 ) +. By combg Thorm, Corollary 4, ad Thorm 3 coclud as follos. Corollary 5 If a -prfct cod xsts J(; ), th a Str systm S(;+;+)ad a Str systm S(;+;0 +) xst. B. Applcatos Lmma 4 mpls that t s suffct toprov that thr ar o-prfct cods J(; ) for. Thrfor, th squl, assum that 0. Assum that a -prfct cod xsts J(; ).By Corollars ad 5, th follog Str systms must xst S(;+;+); S(;+;0 +) S(;+;0 +); S(;+;0 0 +) By Thorm 5, hav that ( + )( +)dvds ( + )( +). ( + )( +)dvds ( 0 + )( 0 +). ( + )( +)dvds ( 0 )( 0 +). ( + )( +)dvds ( 0 0 )( 0 0 +). Sc ( 0 + )( 0 +)0 ( + )( +)( + 3)( 0 ) t follos that ( + )( +)dvds ( + 3)( 0 ) (4) Sc ( 0 0 )( 0 0 +)0 ( 0 )( 0 +) ( 0 + )( 0 ) t follos that ( + )( +)dvds ( 0 + )( 0 ) (5) By Corollary, hav that +dvds 0 ad, thrfor, by (5) hav ( + )( +)dvds ( + 5)( 0 ) (6) Thus, from (4) ad(6) hav ( + )( +)dvds ( 0 ) (7) Thrfor, by Corollary, (4), ad(7), obta th follog thorm. Thorm 4 Assum thr xsts a -prfct cod J(; ). If s odd th s v ad ( + )( +)dvds 0. If s v ad s v th ( + )( +)dvds 0. If s v ad s odd th 0 ( mod 4) ad (+)(+) dvds 0. Corollary 6 Assum thr xsts a -prfct cod J(; ). If s v th ( + )( +)dvds 0. If s odd th 0 (mod 4) ad ( + )( +) dvds 0. Corollary 7 Thr ar oprfct cods J( + p ;), p s a prm ad ; J( + pq; ), p ad q prms, q<p, ad p 6 q 0. C. A Lor Boud o I ths subscto, gv a lor boud o f thr xsts a -prfct cod C J(; ). Ths boud ll b usd our applcato of th ma rsult Scto IV. W assum th xstc of a -prfct cod J(; ) ad as usual 0. Thorm 5 If thr xsts a -prfct cod J(; ), <0, th ( + )( +) > + + f s odd, ad >( + )( +)+ +f s v. Proof W prov th cas of odd. Th cas of v s provd smlarly. By Corollary 6, hav that (+)(+) dvds 0 ad, hc, ( + )( +) 0 By Thorm 3, hav that 0 < 00 (+)(+) < 00. Thus, ( + )( +) > + + ad hc W o hadl th cas of. W dot ++" ad 4 ++", hr " 0 by Corollary 3. W partto th st of coordats N totosubsts A ad B, such that A B, ad thr s a codord from cofgurato (; 0). Lt C() dot th umbr of codords th os th postos of A. No, o ca asly vrfy (s also[6]) that ( ++")!! C( 0 0 ) ( + )!( + ")! C( 0 0 ) C( 0 0 ) ( +) (" 0 ( +)") Sc C( 0 0 ) s obvously ogatv, hav " ( +)" W ot that ">0 or ls th cod s trval. Th Thrfor, hav th follog. " ( +) Thorm 6 If a -prfct cod xsts J(; ),, th +4 +

5 60 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER 004 IV. -REGULAR CODES I ths scto, prst a approach to rul out th xstc of -prfct cods J(; ). W ot that all th o dvsblty codtos hch rul out prfct cods ar drvd from Str systms. I ths scto, vstgat th dvsblty codtos hch ar drvd from th sz of th cod as gv by th sphr-pacg boud. I J(; ), lt us dot 8 (; ) 0 0 th sz of a sphr of radus. Th umbr of codords of a -prfct cod C J(; ) s C 8 (; ) by th sphr-pacg boud, ad hc hav that 8 (; ) (8) Hovr, may do much bttr tha ths by troducg th oto of -rgular cods. Dfto Lt C b a cod J(; ) ad lt A b a subst of th coordat st N f;...; g. For all 0 A df Also, for ach I A df C A () fc C c \A g C A(I) fc C c \A Ig Dfto A cod C J(; ) s sad tob -rgular, f th follog to codtos hold. (c.) Thr xst umbrs (0);...;() such that f A N, A, th C A() () for all 0. (c.) For ay gv -subst A of N, thr xst umbrs A(0);...; A() such that f I Ath C A(I) A(I). Not that f a cod s -rgular,, th t s also ( 0 )-rgular. No, (8) s a smpl rsult of th follog thorm ad th fact that all cods ar trvally 0-rgular. Thorm 7 If a -prfct cod C J(; ) s -rgular, th 8 (; ) 0 0 for all 0. Proof Lt C b a -prfct cod J(; ) hch s -rgular. Lt 0, ad by codto (c.), lt dot th umbr of lgth all-os ords apparg a procto of C oto coordats. W may, thrfor, rt th follog quato, hch couts to dffrt ays th total umbr of lgth all-os ords apparg all th proctos of C oto coordats Thrfor, for ach, 0. 8 (; ) (; ) For th rst of our dscusso, xam -prfct cods J(+ a; ). W df th follog polyomal hch plays a crucal rol (; a; ) (0) a Thorm 8 Lt C b a -prfct cod J( + a; ), ad lt. If (; a; m) 6 0for all th tgrs m, th C s -rgular. Proof W prov th thorm by ducto o. Lt C b a -prfct cod J( + a; ). W partto th coordat st to tosubsts A ad B, such that A ad B + a 0. Th bass for th ducto s. W obta th follog to quatos + a 0 C A (0) C A () 0 C A (0) + C A () + a + + a 0 +a 8 ( + a; ) Th frst quato dscrbs th ay codords of cofgurato (0;) ad (; 0 ) J-covr ords of cofgurato (0;). Th scod quato smply rlats C A (0) ad C A () toth total umbr of codords. To s that ths quato st has xactly o soluto hav tosho that th dtrmat 0 +a a + s ozro. But th dtrmat s smply (; a; ) hch s ozro by th codtos of th thorm. Sc our soluto dos ot dpd o th partto, s mmdatly that th codtos of Dfto ar satsfd. Thrfor th bass s provd. No, for th ducto hypothss, assum that C s ( 0 )-rgular. Hc, thr xst umbrs 0 (0);...; 0 ( 0 ), such that for ach (0)-subst A 0 of N,havC A () 0 (), for all 0 0. W o prov th ducto stp,.., that C s also -rgular. Aga, lt A ad B b a partto of th coordat st N totosubsts, th A ad B + a 0. W start by shog that codto (c.) Dfto for rgularty s satsfd. Ths s do by ducto o th ght of th A part. For ght 0 th clam s obvous. No assum th clam holds for ght,.., ach of th lgth ght ords appars th A part of th codords th sam umbr of tms. W prov that th clam holds for ght +. Lt A 0 A, A 0 0, ad B 0 B, B 0 + a 0 +, b a partto of th coordats hch s obtad from A ad B by movg o coordat from A to B. Wth ths toparttos, fx a lgth 0 ght ord! th A 0 part. Th umbr of codords havg ths ord thr A 0 part s gv by 0 () 0 sc th cod s ( 0 )-rgular. By our last ducto assumpto cocrg ght, th umbr of codords cotag! th A 0 part ad a 0 coordat s gv by C A (). Hc, th umbr of codords cotag! thr A 0 part ad a coordat s th dffrc 0 () 0 (9) 0 CA() W o ot that th choc of coordat has obarg o th last argumts,.., ca us ay coordat of A 0 stad of. Thrfor, th umbr of codords cotag a gv ght +ord th A part s C A ( +). Hc, codto (c.) for rgularty + s satsfd. Aga, ot that (c.) may hold hl (c.) s ot satsfd. I fact, hav provd that f (c.) ad (c.) hold for, th (c.) also holds for +. Thrfor, hav quatos +varabls C A() CA( +) () 0 ; for all 0 0 (0)

6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER Just l th ducto bass, ordr to prov codto (c.) for rgularty add th follog quato m(;) 0 CA() a a 0 () Ths st of quatos has xactly o soluto f ad oly f ts dtrmat s ozro. Ths dtrmat s asly s to b qual to 0 0 (0) a (; a; ) By our assumpto o, hav a uqu soluto to th st of (0) (). Sc th partto dos ot affct th abov argumts, codto (c.) for rgularty holds ad C s -rgular. A. -Prfct Cods W o focus o -prfct cods ad sho that thy ar -rgular for a rlatvly d rag of valus of. Thorm 9 If a -prfct cod xsts J( + a; ), th t s -rgular for all 0 < + a +0 (a +) +4( 0 ) Proof Accordg to Thorm 8, a -prfct cod s -rgular J( + a; ) h (; a; ) 0 ( + a +) + ( + a) + has o tgr roots [;]. Cosdrd as a polyomal, th smallr of th to possbl roots s +a+0p (a+) +4(0), soth rag of dscrbd th thorm cotas o tgr roots. Corollary 8 If a -prfct cod xsts J(; ), + a, th 8 (; ) +( 0 ) 0 0 for all 0 < +a+0p (a+) +4(0). Th follog thorm o bomal coffcts ll b usd to dtrm odvsblty of bomal coffcts by pors of prms. Th thorm as gv by Kummr, ad t ca b foud [, p.45]. Thorm 0 Lt p b a prm. Th umbr of tms p appars a th factorzato of b quals th umbr of carrs h addg b to a 0 b bas p. By [6], alrady o that for -prfct cods, 0 (mod6). Hc, 8 (; ) 0 (mod ). W gv a strogr rsult th follog thorm. Thorm Thr ar o -prfct cods J(; ), h 8 (; ) +( 0 ) 0 (mod 4) Proof Assum thr xsts a -prfct cod J(; ), + a for m m+ 0. W hav th follog to cass. Cas m0. I ths cas 0 m0 < + a +0 (a +) +4(0 ) ; so by Corollary 8 +( 0 ) Thorm 0 mpls that ad so 0 + m0 0 + m0 m0 m0 6 0 (mod 4) +( 0 ) 6 0 (mod 4) Cas m0 0. Not that accordg to Thorm 3, also hav a<0 3. If at to us Corollary 8, hav to sho that 0 ( m 0 ) < + a +0 (a +) +4(0 ) but, aftr rarragg, ths s quvalt toshog that + a + (a +) +4( 0 ) < m+ 0 W o otc th follog + a + (a +) +4( 0 ) < ( 0 ) +4( 0 ) sc a< m+ 0 7 sc m0 0 < m+ 0 as atd to sho. Hc, () holds, ad th by Corollary 8 +( 0 ) Thorm 0 mpls that ad so m m 0 m m (mod 4) +( 0 ) 6 0 (mod 4) () Corollary 9 If thr xsts a -prfct cod J(; ) th thr 0 (mod)or 0 7 (mod ). B. -Prfct Cods, I ths subscto, dscuss otrval -prfct cods h As Scto IV-A, sho that f such a cod xsts, t must b -rgular for a d rag of valus of. Thorm If a -prfct cod,, xsts J( + a; ), th t s -rgular for all 0 < 0. Proof Our am s tosho that (; a; ) 6 0 for all [;0 ) for th rqurd rag of paramtrs (, a, ad ). W actually sho a strogr clam. W sho that s strctly postv th rqurd rag of paramtrs. W start by otg that th polyomal may b rrtt th follog mar by summg a dffrt ordr (; a; ) 0 m(;) 0 (0) a 0 + W cotu ad sho that th r sum, ach of th postv summads s gratr tha ts follog gatv summad absolut valu. Ths s quvalt toshog that a0++ +a0+ <

7 6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER 004 Sc 0,, a 0, ad < Sot suffcs tosho that but ths s quvalt to hch alays holds. +a0++ +a0+ ( 0 )( 0 )( + a 0 + +) ( + )( 0 )( + a ) < ( 0 )( ) ( 0 + ) ( 0 )( ) ( 0 + ) ( 0 ) + ( +) 0 ; Corollary 0 If a -prfct cod xsts J(; ), th t s -rgular. Proof Assum thr xsts a -prfct cod J(; ) By Thorms 5 ad 6, hav that > ad by Thorm such a cod s -rgular for all < 0, ad hc th cod s -rgular. I th xt thorm xtd th rag of rgularty gv Thorm. W us Corollary 0 as th startg pot for th proof. Th mthod usd th proof of Thorm o logr ors for th xtdd rag, so a asymptotc approach s usd. W start by gvg tosmpl ll-o dtts, hch ca b provd by basc combatoral tchqus. Lmma 5 0 p m p 0 (0) 0 m 0 Lmma 6 Vadrmod s covoluto m p 0 0 p m 0 p p Thorm 3 For all, thr xsts W > 0 such that for all W, all -prfct cods J( + a; ) ar -rgular. Proof Our proof starts sstally th sam as th proof of Thorm. W actually at to sho, that for a larg ough, th a 0 ad (; a; ) 0 > 0 m(;) 0 (0) a 0 + By Corollary 0, may cosdr, so hav tosho that (0) Th lft-had sd ca b rrtt as 0 + a 0 (0) 0 + a > 0 +a0+ +a0 W cotu by provg that for all 0, th r sum s postv,.., No (0) 0 0 v 0 0 (0) a0+ +a a0+ +a odd +a0+ +a0 +a0+ +a0 0 0 odd odd hr th last stp s ta by usg Lmma 5. Soo t s ough to prov that +a0+ +a0 0 0 odd 0 0 < 0 W ot that th sum may b rrtt th follog mar 0 odd (0) Pluggg ths to(3) hav toprov that, +a0+ +a Fally, hav th follog cha of qualts +a0+ +a a a a a `0 0 ` + a a (3) by Lmma < (4) 0 (5) 0 00` (6) (0) a0+ +a0 > `0 0 ` ` 0 (7)

8 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER a a a a (8) 0 (9) hr th trasto from (5) to (6) s by Lmma 6, th trasto from (7) to (8) s by Nto s bomal dtty, ad th trasto from (8) to (9) s by usg a 0,, ad. Thrfor, t s ough that sho that For a fxd valu of lm! ad hc, a W xsts as rqurd < (0) 0 0 W ot that Thorm 3 may b asly xtdd to sho that for all ad 0 < <, thr xsts W ; > 0 such that for all W ;, all -prfct cods J( + a; ) ar bc-rgular. Hovr, for th follog, s suffct. Thorm 4 Thr ar o -prfct cods J(; ),, hch ar also bc-rgular, h 8 (; ) 0(mod4). Proof Lt C b a bc-rgular -prfct cod J(; ), + a,for m m+ 0. W dstgush bt tocass. Cas m0. I ths cas 0 m0 Sc th cod s bc-rgular, th by Thorm 7 8 (; ) Thorm 0 mpls that ad so 0 + m0 0 + m0 m0 m0 6 0 (mod 4) 8 (; ) 6 0 (mod 4) Cas m0 0. Not that by Thorm 3, also hav a< 0 ( +) < If at tous Thorm 7, hav tosho that But o 0 ( m 0 ) () 0 + a 0 < <m 0 Hc, () holds, ad th by Thorm 7 8 (; ) Thorm 0 mpls that ad so m m 0 m m (mod 4) 8 (; ) 6 0 (mod 4) Thorm 5 Thr ar o -prfct cods J(; ),, hch ar also bc-rgular, h 8 (; ) 0 (mod p ), p 3 a prm. Proof Lt C b a -prfct cod J(; ), forp m p m+ 0. No,f p m0 0, th hav < p hch s mpossbl for p 3 by Thorm 3. Hc, lt p m0 ( +)p m0 0, for som p 0. I ths cas 0 p m0 Sc th cod s bc-rgular, th by Thorm 7 8 (; ) Thorm 0 mpls that ad so 0 + p m0 0 + p m0 p m0 p m0 6 0 (mod p ) 8 (; ) 6 0 (mod p ) Corollary Thr ar o -prfct cods J(; ),, hch ar also bc-rgular, h 8 (; ) 0 (mod p ), p a prm. To prov th xt thorm d aothr trstg thorm o bomal coffcts. Ths thorm s du to Lucas []. Lt a 0 b som tgr. W th dot by p (a; ), th th dgt of a h rtt bas p. Hc, a 0 p(a; )p Thorm 6 Lt p b a prm, ad m 0 totgrs, th m 0 p(; ) p(m; ) (mod p) Thorm 7 Lt p b a prm, ad 0 (mod p ).Ifa-prfct cod xsts J(; ), th 8 (; ) 0 (mod p ) Proof Lt C b a -prfct cod J(; ). By Corollary, (mod +)ad hc + 0 +

9 64 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER 004 0(modp ). I othr ords, th to last sgfcat dgts th rprstato bas p of,, ad 0, ar both p 0,.., p(; 0) p(; ) p( 0 ; 0) p ( 0 ; ) p (; 0) p (; ) p 0 () Lt 0 <b som tgr such that 0 (mod p).no p0 p 0 + p p by (4) 0 (mod p ) Fally, usg th fact that modulo p quals 0 + p by (3) 0 (mod p ), th sphr sz ( 0 )( 0 0 ) ( +) Hovr, ot that 0, 0 0, ad +ar coprm to p. Furthrmor, ( + ) (mod p ). Hc, 0 Ths may b rpatd togt (mod p ) 0 8 (; ) 0 p 0 0< 0 0 (modp ) 0 (mod p ) p p 0 (mod p ) (3) No lt 0 < b som tgr such that 0 (mod p ). Not that all th umbrs of th form + p, h 0 p 0, oly th scod dgt bas p chags hl th frst dgt s alays zro. W xam th follog sum modulo p usg Thorm 6 p0 0 + p p0 0 `0 p p p 0 ` (p 0 ) p 0 ` p(; `) p( + p; `) p(; `) p( + p; `) p(; `) p( + p; `) p( 0 ; `) p( + p; `) p( 0 ; `) p( + p; `) p( 0 ; `) p( + p; `) (mod p) Hovr, p((p 0 ); 0) p 0 <p0 p (p 0 ; 0), ad thrfor, by Thorm 0 (p 0 ) p 0 0 (mod p) Hc, th prvous sum s cogrut to 0 modulo p. No, for som tgr p0 0 + p 0 + p p (4) W cotu by xamg th follog sum modulo p p p0 p0 ` p + ` 0 + p + ` Corollary For ay gv, 0 (mod p ), p prm, thr ar ftly may otrval -prfct cods th Johso graph. V. APPLICATIONS A smpl obsrvato s that th lft-had sd of (0) s a mootoously dcrasg fucto. Hc, a smpl computr sarch ca fd th valu of W of Thorm 3 ad valdat that (; a; ) has o tgr roots for ad W. Such a computr sarch as do for 3; 7; 8 ad, dd, o such roots r foud. Thrfor, coclud th follog. Proposto Thr ar ootrval 3-prfct, 7-prfct, ad 8-prfct cods th Johso graph. Aothr computr sarch as coductd hch tstd th dvsblty codtos of Thorm 7. Th rsults of ths sarch ar gv th xt to propostos. Proposto Thr ar o -prfct cods J(; ) for all Proposto 3 Thr ar o -prfct cods J(; ) for all Ths s a sgfcat mprovmt ovr th prvous mthod usg Str systms, hch lft for all 0 (mod 6) as caddats, ad for all 0 ; 6; 50 (mod 60) as caddats. W blv that furthr umbr-thortc aalyss of th rgularty mthod ll rul out all prfct cods. Fally, for gv ad a, xam hch graphs J( + a; ) th xstc of -prfct cods as ot ruld out. Th rsults of Sctos III ad IV ad carful aalyss sho th follog. Thorm 8 For a 35, thr ar o -prfct cods J( + a; ) th th follog possbl xcptos -prfct cods ad -prfct cods J(+;) ad J(+4;), ad 4-prfct cods J( +5;) ad J( +30;). VI. CONCLUSION Th ma purpos of ths corrspodc as to attac Dlsart s 30 yars old coctur o th oxstc of otrval prfct cods th Johso schm. W shod varous rsults hch rul out -prfct cods J(; ) for varous valus of,, ad. A ovl tchqu usg -rgular cods as troducd. For practcal us, ths tchqu s abl to rul out ay gv st of paramtrs.

10 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER Th ma problm that should b th focus of furthr rsarch s to prov that thr ar o otrval prfct cods th Johso schm by usg th cocpt of -rgular cods. Cocyclc Smplx Cods of Typ Ovr ad Nmalsr Paala ad Asha Rao ACKNOWLEDGMENT Th authors ould l to tha th to aoymous rfrs hos commts hav amdd ths corrspodc. REFERENCES [] F. J. MacWllams ad N. J. A. Sloa, Th Thory of Error-Corrctg Cods. Amstrdam, Th Nthrlads North-Hollad, 978. [] J. H. va Lt, Noxstc thorms for prfct rror-corrctg cods, Computrs Algbra ad Numbr Thory, vol. IV, SIAM-AMS Procdgs, 97. [3] A. Ttävä, O th oxstc of prfct cods ovr ft flds, SIAM J. Appl. Math., vol. 4, pp , 973. [4] V. A. Zovv ad V. K. Lotv, Th oxstc of prfct cods ovr Galos flds, Probl. Cotrol ad Iform. Thory, vol., pp. 3 3, 973. [5] M. R. Bst, Prfct cods hardly xst, IEEE Tras. Iform. Thory, vol. IT-9, pp , May 983. [6] Q. A. Nguy, L. Györf, ad J. L. Massy, Costructos of bary costat-ght cyclc cods ad cyclcally prmutabl cods, IEEE Tras. Iform. Thory, vol. 38, pp , May 99. [7] A. E. Brour, J. B. Sharr, N. J. A. Sloa, ad W. D. Smth, A tabl of costat-ght cods, IEEE Tras. Iform. Thory, vol. 36, pp , Nov [8] F. R. K. Chug, J. A. Salh, ad V. K. W, Optcal orthogoal cods Dsg, aalyss, ad applcatos, IEEE Tras. Iform. Thory, vol. 35, pp , May 989. [9] T. Etzo, Costructos of rror-corrctg DC-fr bloc cods, IEEE Tras. Iform. Thory, vol. 36, pp , July 990. [0] H. C. A. va Tlborg ad M. Blaum, O rror-corrctg balacd cods, IEEE Tras. Iform. Thory, vol. 35, pp , Spt [] P. Dlsart, A algbrac approach to assocato schms of codg thory, Phlps J. Rs., vol. 0, pp. 97, 973. [] E. Bggs, Prfct cods graphs, J. Comb. Thory Sr. B, vol. 5, pp , 973. [3] E. Baa, Cods b-partt dstac-rgular graphs, J. Lodo Math. Soc., vol., pp. 97 0, 977. [4] P. Hammod, O th oxstc of prfct cods ad arly prfct cods, Dscr. Math., vol. 39, pp , 98. [5] C. Roos, A ot o th xstc of prfct costat ght cods, Dscr. Math., vol. 47, pp. 3, 983. [6] T. Etzo, O th oxstc of prfct cods th Johso schm, SIAM J. Dscr. Math., vol. 9, o., pp. 0 09, May 996. [7], O prfct cods th Johso schm, DIMACS Srs Dscrt Mathmatcs ad Thortcal Computr Scc, vol. 56, pp. 5 30, 00. [8] W. J. Mart, Compltly rgular substs, Ph.D. dssrtato, Uv. Watrloo, Watrloo, ON, Caada, 99. [9] O. Shmabuuro, O th oxstc of prfct cods J( +p ;), Ars Combatora, tob publshd. [0] R. Ahlsd, H. K. Ayda, ad L. H. Khachatra, O prfct cods ad rlatd cocpts, Ds., Cods Cryptogr., vol., o. 3, pp. 37, Ja. 00. [] R. L. Graham, D. E. Kuth, ad O. Patash, Cocrt Mathmatcs A Foudato for Computr Scc. Radg, MA Addso-Wsly, 994. [] N. J. F, Bomal coffcts modulo a prm, Amr. Math. Mothly, vol. 54, pp , 947. Abstract Ovr th past dcad, cocyls hav b usd to costruct Hadamard ad gralzd Hadamard matrcs. Ths, tur, has ld to th costructo of cods slf-dual ad othrs. Hr xplor ths das furthr to costruct cocylc complx ad Butso Hadamard matrcs, ad subsqutly us th matrcs to costruct smplx cods of typ ovr Z ad Z, rspctvly. Idx Trms Butso, cocycl, complx Hadamard, xpot, quatrary, slf-orthogoal, smplx cods, trac. I. INTRODUCTION Varous authors [], [], [], [] hav studd th costructo of cocyclc Hadamard ad cocyclc gralzd Hadamard matrcs ad th us of ths matrcs th costructo of cocyclc cods. Hr xtd ths costructos to obta cocyclc Butso ad cocyclc complx Hadamard matrcs. Smplx cods of typ r studd by Gupta [9], but o mthods of costructos r gv. W us th cocyclc complx ad cocyclc Butso Hadamard matrcs to costruct smplx cods of typ ovr Z 4 ad Z, rspctvly. W assum that th radr s famlar th th basc facts of th thory of Hadamard matrcs (s, for xampl, [5]) ad of bary lar cods (s [3]). If G s a ft group (rtt multplcatvly th dtty ) ad C s a Abla group, a cocycl (ovr G) s a st mappg GG!C hch satsfs (a; b) (ab; c) (a; bc) (b; c); 8a; b; c G A cocycl s ormalzd f (; ). A cocycl may b rprstd as a cocyclc matrx M [ (a; b)] a;bg oc a dxg of th lmts of G has b chos. Lt C p b th multplcatv group of all complx pth roots of uty, C p f;x;x ;...; x p0 g, hr x xp(p) ad p s a tgr. A squar matrx H [h ] of ordr th lmts from C p s calld a Butso Hadamard matrx (BH(; p)) (s [5]) f ad oly f HH 3 I, H 3 bg th cougat traspos of H ad I th dtty matrx of ordr. Wh p ad ; or a multpl of 4, BH(; p) s a Hadamard matrx. A complx Hadamard matrx H of ordr s a matrx th trs from f;;0; 0g that satsfs HH 3 I, hr p 0 ad H 3 s th cougat traspos of H. It s cocturd that a complx Hadamard matrx xsts for vry v ordr. I [5], t s sho that vry complx Hadamard matrx has ordr or dvsbl by. A complx Hadamard matrx s a spcal cas of a Butso Hadamard matrx BH(; p) for p 4. Lt H [h ; ] b a squar matrx ovr C p, hr p s a fxd tgr p >. Th matrx E [ ;]; ; Z p, hch s obtad from H [x ][h ; ], hr x xp(p), s calld th xpot matrx assocatd th H. Th lmts of th xpot matrx E l th Galos rg GR (p; ) (Galos fld GF (p), forp prm), ad ts ro vctors ca b vd as th codords of a cod ovr th tgrs modulo p /04$ IEEE Mauscrpt rcvd Dcmbr 0, 003; rvsd May 5, 004. Th authors ar th th Dpartmt of Mathmatcs ad Statstcs, Royal Mlbour Isttut of Tchology, GPO Box 476V, Mlbour, VIC 300, Australa. Commucatd by C. Carlt, Assocat Edtor for Codg Thory. Dgtal Obct Idtfr 0.09/TIT