TiCC TR November, Gauss Sums, Partitions and Constant-Value Codes. A.J. van Zanten. TiCC, Tilburg University Tilburg, The Netherlands

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1 Tlburg ceter for Cogto d Coucto P.O. Box 953 Tlburg Uversty 5 LE Tlburg, The Netherlds htt:// El: tcc@uvt.l Coyrght A.J. v Zte, 3. TCC TR 3- Noveber, 3 Guss Sus, Prttos d Costt-Vlue Codes A.J. v Zte TCC, Tlburg Uversty Tlburg, The Netherlds d V.V. Vvrek Isttute of Mthetcs d Ifortcs, Bulgr Acdey of Sceces Trovo, Bulgr

2 Abstrct We study the reltosh betwee rttos of soe teger GF() dstct rts of sze t ost ( ) /, d bry vectors : (,,..., ( )/) of vlue defed s : ( )/ j j. I rtculr we vestgte grou of trsfortos ctg o the j fly A { A, A,..., A } where A stds for the set of ll vectors of vlue. By exlotg roertes of ths reltosh, we re ble to detere the sg of the / qudrtc Guss su G() : e. I the lst secto of ths reort we ly the the develoed theory to derve soe reltos for cert subsequeces of the Morse- Thue sequece.

3 Cotets. Itroducto. 4. Guss sus d rttos Deftos d relres A grou of trsfortos. 5. Exles for = 7,, 3 d Costructg A fro A. 7. Deterg the sg of the Guss su G () Orderg the vectors of ( )/ GF () two dfferet wys The Morse-Thue sequece d relted roertes. 37 Refereces. 5 3

4 . Itroducto As s well kow, the followg equlty holds for y teger exceedg oe or, equvletly / e, () G () : e () / whch c be cosdered s the su of fte geoetrc seres. I ths reort we shll lwys ssue tht s odd re. A less eleetry roble s the suto of / / G () : ( ) e ( ) e, (3) where ( ) s the Legedre sybol, beg equl to f s squre GF () d equl to f s osqure. It wll be cler tht ( ) c be terreted s chrcter of the ultlctve grou of GF() whe rus through ll ozero eleets of the feld. Sce ths chrcter s of order, d sce t ws Guss who set hself the tsk to detere the su (3), ths su s usully clled G (). Slrly, the su () s clled G (), where we y thk the trvl chrcter ut frot of the e - ower. More geerlly, oe defes Guss su / G( k) : ( ) e, (4) 4

5 where stds for chrcter of the grou GF ( ) of order k. Here d the ext, we dot the coveto tht f o bouds re dcted, the suto vrble rus fro utl. Eve ore geerlly, oe defes Guss sus wth resect to feld F : GF( q), q s * G (, ) : ( c) ( c), (5) c F \{} where s ultlctve chrcter of F d ddtve chrcter. More recsely, for d we c substtute j k jk / q ( g ) e, k {,,..., q }, (6) where g s geertor of GF( q ) d b Tr ( bc) / q ( c) e, b, c F. (7) Oe c see the Guss su (5) s relto coectg the chrcters of the two bel grous of fte feld. Fro ow o we oly shll del wth the qudrtc Guss su G (). Oe c esly derve (cf. e.g. Secto ) tht G () ( ), (8) Ad so G() =. Moreover, by relcg (3) the suto vrble by oe c lso show tht ±, f od 4, G () { (9) ±, 3 od 4. These reltos lso follow edtely fro (8), sce ( ) equls for od 4 d for 3 od 4. I My 8 Guss cojectured tht the true sgs (9) re the lus sgs. However, t took h other four yers to rove ths cojecture. O August 3, 85, he wrote hs dry: At legth we cheved deostrto of the very elegt theore etoed before My, 8, whch we hd sought for ore th four yers wth ll efforts. 5

6 Sce the, the bove roble s kow s the deterto of the qudrtc Guss su or better the deterto of the sg of the qudrtc Guss su. My dfferet roofs hve bee foud, s well s coectos to other thetcl robles of vrous ture. For revew we refer the reder to []. I the reg rt of ths reort we shll vestgte the reltosh whch ers to exst betwee ths roble d the theory of rttos d lgebrc codg theory.. Guss sus d rttos We cosder g the qudrtc Guss su G() (3) whch we ow wrte s d we lso troduce the roduct S ( ) ( ), e /, () ( ) / P ( ) ( ), The followg le c esly be derved.. () LEMMA For y odd re, d for ll b {,,, } oe hs () () S P ( ; (. b ( ) ) b ( ) ) Proof: The roofs follow by strghtforwrd clcultos. For the ske of coveece, we oly gve the roofs for b. () Frst we wrte obt S ( ) S l Q j j j k k / ( ) ( ) ( ) ( ). By usg l l Q l, k l : k /, we, where Q s the set of qudrtc resdues GF( ). Now, the ddtve chrcters of GF() stsfy the orthogolty reltos ( l ), f l, d ( ), f l. Alyg tht Q l 6

7 ( )/ d dstgushg betwee (= od ) Q d the result, whch holds for d, resectvely. Q, yelds () Sce s rtve -th root of uty, we hve tht ( ) / P ( ) { ( )} ( ) = (... ) ( ) / 4 ( ) / ( ) ( ) ( ) ( ) ( x... x ) x = for j,,, ( ) /. Hece, ( ), sce ( ) ( )/ d. We lso used the fctorzto ( x )( x )...( x ) x. j j j j Next we relce the lgebrc uber the exressos () d () of Le d lso () d (), by vrble x, d we forulte the followg roerty. THEOREM Let be soe odd re. The there s {, } wth ( ) / ( x ) x ( ) /8 ( ) x, od x. Proof. We wrte the equlty the Theore s Q( x) R( x), od x, where Q (x) d R (x) re cosdered s olyols Q[x]. Fro Le, t follows tht the ubers b b / e, b {,,, }, re ll zeros of Q ( x) R( x). For the odd vlues of b, these ubers re lso zeros of x. So, whe crryg out the dvso lgorth d wrtg Q( x) R( x) q( x)( x ) r( x), t follows tht r(x) hs t lest zeros. But the degree of r (x) s t ost. Therefore r (x) s the zero olyol. Furtherore, we hve x ( x )( x x... ). Sce the secod olyol the rhs s rreducble over Q for odd res, t s coted ether Q ( x) R( x) or Q ( x) R( x) or both. It s lso obvous tht these two olyols both hve s zero. So, t lest oe of the two cots x x... whe fctorzed. Hece, t lest oe of the two s detcl to zero. 7

8 Now f both were, t would follow tht Q (x) d R (x) both were equl to zero od x whch s flse. We y coclude tht for ech odd re, there exsts uque sg s dcted the Theore. As cosequece t follows whe substtutg for x tht P ) S ( ). () ( We lso hve fro defto () d Le () tht P (3) ( )/ ( ), becuse I ( ) > d Re ( ) for {,,., ( ) / }. So,, od 8,, 5 od 8, S ) { (4) (, 3 od 8,, 7 od 8. The roble of the sg of the qudrtc Guss su turs out to be equvlet to the deterto of the sg the olyol equlty of Theore. To ths ed we core the coeffcet c of the rhs ths coeffcet s equl to x, for soe teger, both sdes of the equlty. I ( ), where GF ( ) s defed by ( ) /8. (5) To detere c the lhs, we cosder the N rttos of to uequl rts of sze t ost ( ) /. More recsely, we study the equto... k od, (6) k, such tht ll, ( ) /, re dfferet. If N e d uber of the rttos wth eve k d odd k, resectvely, the ( ) ( ) / ( N e N o N o deote the c equls ). The study of such rttos wll be the subject of the ext secto. 8

9 3 Deftos d relres Let be soe odd re. We shll study the rttos of ostve tegers cosstg of uequl rts the sze of whch s t ost /. It wll be obvous tht we c rereset such rttos by bry vectors c c, c,..., c ) of legth ( ) /. ( ( ) / Here, c = f d oly f the rtto cots rt of sze. We terret ll vectors s row vectors. The uber of oes such vector c s clled the weght of the rtto d s deoted by c. It stds for the uber of rts the rtto. Let c be soe rtto. We defe ( ) / jc j od (7) j d cll the vlue of c or vl (c ), wth {,,..., }. For fxed vlue, we collect ll vectors hvg ths vlue set A cosstg of A bry vectors of legth ( ) /. So, ths set cots ll covetol rttos of the tegers,,,... to uequl rts of sze t ost ( ) /. We shll cll such set costt-vlue code odulo or brefly costt-vlue code. Corresodg to the lst les of the revous secto, we troduce tegers N d N, beg the uber of vectors A wth eve uber of oes d odd uber, resectvely. The coleet of rtto c s defed s the rtto corresodg to the vector c c = c +, where s the ll-oe bry vector of legth ( ) / Sce the vlue of s equl to ll vectors of set c e L : ( ) / 8 od, (8) A hve coleet of the se vlue c L o. Hece, we c wrte A AL, d we cll A the coleet of A. We lso eed the vlue of the frst hlf of, defed by K... [( ) / 4] ( ± 3) /3 od, (9) for ± od 4. Cosequetly we hve L 4 K ( ) / 4 od. () 9

10 Furtherore, fro deftos (5) d (8) t follows tht L ( ) /8 () s equlty GF (). I order to del wth the sets A, {,,..., }, we lso troduce the teger ( ) /, ±3 od 8, N ( ) { () ( ) /, ± od A grou of trsfortos Let I {,,.. ( ) / } d let be soe teger wth. For ech such, we troduce dex sets : { :, I I I I }, I I : I \ I, (3) where the roduct s tke od. As s lredy dcted (3), we shll ot soetes the reter, whe there wll ot rse cofuso whch vlue hs The su of the tegers I, resectvely I, wll lso ly role the ext. We defe both tke od. S :, I S := ' I, (4) The sets I d I re closely relted to Guss crtero for the vlue of ( ), GF(), beg f s squre d f s ot squre GF (). The followg le wll llustrte ths. LEMMA () ( ) ( ), wth I ;

11 () ' S S ; () for ll o-zero GF ( ) oe hs S d ' S. Proof. () Let I {,,...} d I { b, b,...}. For y I, the uerclly lest resdue od s equl to, wheres for y b I ths vlue equls b. So equlty () s recsely Guss crtero (cf. e.g. [3,. 59]). () Ths relto follows edtely fro the observto tht the set I for s detcl to the set I for. () It c esly be roved (cf. [ 3,.59}) tht the ubers,,, b, b,... re erutto of,,., ( ) /. Hece, we hve od the equlty ' ( S S ) (...) ( b b...)... ( ) / Furtherore, edtely. ( S S ) ' I L. L. The two reltos the Le ow follow Next, we defe for ech GF() * erutto trx P of sze I x I wth eleets, j, j od, I, or j od, I, { (5) otherwse., j Art fro cses where we eed the -deedece of P exlctly, we shll ot the dex the ext. THEOREM Let l be the order of od,.e. ord reresets erutto o I cosstg of of ( ) / l cycles of legth l, for l odd. ( l. The the trx P defed by (5) ) ( ) / l cycles of legth l /, for l eve, d Proof. Cosder the g : GF ( ) GF( ), ( ). Ths g, whch erutes the eleets of GF (), c be odfed to erutto of the eleets of I the followg wy. Frst, erutes the ozero eleets of GF () ccordg to ( ) /l cycles of legth l. Next, we chge ll eleets these cycles whch re ot I to ':, d the we ot the us sg of '. If s the se cycle s, whch s the cse f l s eve, ths cycle of legth l s trsfored to cycle

12 of legth l / followed by the se cycle of legth l /, whle ll eleets ow re I. The se holds for ll other cycles. If d re dfferet cycles of legth l, whch s the cse f l s odd, the both cycles becoe detcl fter chgg the us sgs. So, whe ottg reeted cycles, we ed u wth erutto of the eleets of I s descrbed the theore. For the g the se holds. More recsely, the trx P reresets the g, odfed by the bove rocedure. We defe the followg trslto vector ( ) / GF () for every GF( )* t = (t, t,., t ), t ff j od, I. (6) ( ) / j Aother wy to chrcterze the vector t s by syg tht t j f d oly f j I, where I stds for the set { : I}. Slrly, we c sy tht f the trx eleet, the ether j I or j I. Notce tht the followg reltos hold od, j I I I, I I I, (7) where stds for dsjuct uo. The trces P d the vectors t stsfy the followg eleetry roertes whch wll be used the theory to be develoed the reg rt of ths reort. LEMMA 3 () P P for ll GF() * (, j d for ll Z. () Whe P ), the j f the set {,,,..., } cots eve uber of eleets fro I, whle eleets fro I. () t = t +. j f ths set cots odd uber of Proof. () As cosequece of defto (5), we hve tht the, j - eleets of both trces re equl to f d oly f j ±. () Ths stteet s obvous fro the deftos of the trx P d the set I ; () The sets I d I corresodg to re equl to the sets I d I resectvely, corresodg to.

13 Lke the cse of the trx P, we shll ofte ot the dex d just wrte t whe t s obvous wht -vlue s et. Next, we cosder for soe fxed -vlue, the trsforto ( ) / ( ) / T : GF() GF() defed by T ( c ) = c P + t. (8) Sce P s vertble, so s T, d hece t erutes the vectors of ( ) / GF (). THEOREM 3 Let be geertor of GF ()*, the for ll c ( ) / ( ) / ( ) / ( )/ GF( ). () T ( c ) c c for od 4, d T ( c) = c for 3 od 4; () the order of T s erutto of d to ( ) / for 3 od 4. GF () s equl to for od 4, Proof. For the ske of coveece, we dot ths roof the otto l :, beg the order of. () For rbtrry vlue of we hve T ( c) = c P + t ( P... P E). (9) Accordg to Theore, the erutto trx P reresets oe cycle d of legth l/, l / whch we wrte s d (d d. d ), I. Hece, c P c. I order to l / / detere the secod ter the rhs of (9) for l /,.e. t :=t ( P l... P E), we rerk tht for ths -vlue, the trx betwee retheses s equl to the ll-oe trx. For ech od l /, we hve ether d d or d d od. Let d r l / r l cot r eleets fro I, the d ( ) d ( ) d, sce, d so r ust be odd (cf. Le 3 ()). If od 4, we hve tht l / s eve. It follows tht d cots odd uber of eleets fro I. Fro deftos (3) d (6) t ow d l / T follows tht t cots odd uber of oes. We coclude tht t =, d hece ( c) = c + = c c. If 3 od 4, the l / s eve, d t follows slrly tht d cots eve uber of eleets fro I d tht t cots eve uber of oes. l / So, t =, d hece, T ( c) c + = c. 3

14 T () Let od 4. We kow lredy tht I, the detty oertor. Assue tht T (c) = c for ll c ( ) / GF ( ), for soe wth. It the follows P fro (9) tht c ( P E) = t(... P E) for ll vectors c. Multlyg ext by P E rovdes us wth (c + t) ( P E) =, for ll c, where c = c ( P E). Ths c oly be true f P E s the zero trx, d so l /. But ths cotrdcts the frst relto (). Hece, the order of T s equl to ths cse. For 3 od 4, the roof s coletely slr. THEOREM 4 Let be soe eleet of GF ()* d let be ts verse, the T = T. Proof. If I {, b,...} s the dex set corresodg to, the I ' {, b,...} s the dex set corresodg to, s wll be cler fro the defto of I (3). If P d t re the erutto trx d trslto vector w.r.t. d f P ' d t std for the se otos w.r.t., the t wll be obvous fro the deftos (5) T T d (6) tht P' P P d t = t P t P. Hece, T T ( c) = c PP ' + t P ' + t = c PP + t P + t P = c, for ll c. THEOREM 5 For ech, A,,, T duces erutto o the fly of sets A A, A } such tht ( A ) Ab, wth b ( S ) d S Proof. Let A wth vlue. Frst, we shll detere the vlue b of the vector b = T ( ). The cooets wth I cotrbute ( ) to b, whle those I yeld ( ). Hece, both cotrbutos together d tke od, gve b = I I = S corresodece betwee the vectors of. Alyg Theore 4 ow gves tht there s oe-oe I A d those of b I. { I A wth b ( S ). Exles I {,,., [( ) / 4] }, I I \ I, t = (,,,, ), b = ( S ) ( K ) ; 4

15 ( ) / I = {, 3, 5,..}, I = {, 4, 6, }, t = (,,,,,,,,.,), b = ( ) / (S ( ) / ) = ( ) / ( L K ) ; I, I I, t =, P E. b =. I I, I, t =, P E, b S L. The eruttos,, geerte erutto grou G A o the fly A of costt-vlue codes. I the reg rt of ths secto we shll vestgte ths grou soewht closer. Let w, be the vlue of the set A the tegers w, stsfy GF () the recurrece relto ( ), for fxed d. Accordg to Theore 5, w, ( S w, ), w,, (3) Ths recurrece relto s ler d c esly be solved, yeldg, ( ( ) ) ( = w ) ( )( ). (3) Here, we lso led Le (). THEOREM 6 () for y GF() oe hs for odd d for eve; () G A hs oe orbt {A } of sze, wheres ll other A,, re oe orbt of sze ; * () the erutto geertes G A f d oly f s geertor of GF ( ) ; (v) for ll oe hs A = N (), wheres A = N ( ) for ± od 8, d A = N () for ±3 od 8. Proof. Prt () follows esly by lyg (3) wth d relcg by odd cse, d by the eve cse. Geerlly, the erutto geertes subgrou of equto w, s equvlet to the G A. Fro (3) we hve tht the 5

16 ( )( ( ) ). (3) Let l be the order of od. The oly -vlue whch stsfes eq. (3) for soe fxed wth < < l, s. So, A s vrt wth resect to ll trsfortos of G A. Furtherore, t wll be cler fro (3), tht the legth of the orbt to whch A,, belogs uder the cto of, s equl to l. So, f we tke, where s geertor of GF( )*, the orbt hs legth.prt (v) follows edtely fro the fct (cf. Theore 7) tht ll A,, hve the se sze, whle A = A. We rerk tht Theore 6() wth s lso edte cosequece of Theore Exles for = 7,, 3 d 7 I ths secto we reset exles for the cses = 7,, 3 d 7 whch rereset ll ossbltes od 8. () For = 7, we hve I {,,3}, L 6, 3 d K. I ths cse geertes the grou GF (7)*, wheres does ot. For we obt I {,3}, I { } d P, t = (,, ). 3 Hece, T ( c) = c P 3 + t ( P P E) = c + = c, whch llustrtes Theore 3 (). The costt-vlue codes re A {(,,)}, A {(,, )}, A {(,, )}, A {(,,), (,,)}, 3 A {(,,)}, A {(,, )}, A 6 {(,,)}. 4 5 Sce s geertor of GF (7)*, the trsforto geertes the grou G A by Theore 6, d t cts trstvely o the fly { A }. For oe hs I { }, ' I {,3}, d so S d S 5. The trsforto (A ) Ab wth b ( ) od 7 gves rse to the orbts 6

17 A, A A A5 A6 A4 A A, 3 A 3 wheres (A ) A b, wth ( b 5 ) od 7, roduces the orbts A, A A4 A5, A A6 A A 3 A 3 () For we hve the followg dt: L 4,, K 3. I ths cse s geertor of the grou I {, } whch gves rse to * GF (), d s ot. For we obt P, d t = (,,,, ). It follows edtely tht (cf. lso Theore 3() d ts roof) 5 T ( c) = c P t ( P... P E) = c E + t J = c + = c. The fly A of costt-vlue codes cossts of the sets: A {(,,,,), (,,,,), (,,,,)}, A {(,,,,), (,,,,), (,,,, )}, A {(,,,,), (,,,,)}, A {(,,,,), (,,,,), (,,,,)}, A {(,,,,), (,,,,), (,,,, )}, 3 A {(,,,,), (,,,,), (,,,,)}, A {(,,,,), (,,,,), (,,,, )}, 5 A {(,,,,), (,,,,), (,,,,)}, A {(,,,,), (,,,,), (,,,, )}, 7 A {(,,,,), (,,,,), (,,,,)}, A {(,,,,), (,,,,), (,,,, )} Sce s geertor of GF (), the trsforto 9 s geertor of G A ccordg to Theore 6, d t cts trstvely o the fly { }. I order to ly Theore 5, we obt for 9 tht I {3, 4, 5}, d hece S A 9 7

18 od. Ideed, the reltos trsfortos: (A ) A 9 b d b 9( ), od, rovde us wth the A A, A A9 A5 A8 A3 A4 A6 A A7 A A. * Sce does ot geerte GF (), we cot ly Theore 6 () for. For, we hve I {, } d S 3. Whe lyg (A ) Ab, b (3 ) od, we obt A A, A6 A5 A7 A3 A A, A A4 A9 A A8 A. So, does ot ct trstvely o the fly { }. It lso follows fro the bove sequece of trstos tht the grou eleet orbts A hs A A, A 5 A3 A6 A7 A A, A A9 A8 A4 A A We core these orbts wth those of 7. For 7, we fd I {,5 }, d so S 7 d b 7(7 ). The ltter relto gves the se sequeces of dces of the 7 sets A s dd, thus llustrtg the eve cse of Theore 6 (). The odd cse s 3 deostrted by corg the trsfortos d 3 8. For 8 we fd I {,3}, S 5 d b 8(5 ). Ths lst equlty yelds the followg orbts wth resect to 8 : 8 A A, A7 A6 A3 A5 A A, A A A4 A8 A9 A As oe c esly verfy, the trsforto 3 gves rse to the se orbts. As llustrto of Theore 4, we tke 9 5 od. Now, we fd I {,3,5 } d S 5 9. By lyg b 5(9 ), we derve tht 5 erutes the sets of A reversed order s 9 dd : A A A7 A A6 A4 A3 A8 A5 A9 A Slrly, we derve for 6 od, tht I {,4}, S 6 6 d ext by lyg b 6(6 ) the erutto A A3 A7 A5 A6 A, A A8 A A9 A4 A 8

19 6. whch shows tht () For 3 we fd L 8, 4, K 6 * Both d re geertors of GF (3). For we hve I {,, 3} d I { 4, 5, 6}, fro whch we fer P, d t = (,,,,, ) So, T ( c) = c P + t ( P... P E) = c E + t J = c + = c c (cf. Theore 3). For we hve I {4, 5, 6} d I {,, 3}, d hece P, d t = (,,,,, ). 6 Ag t follows tht T ( c) = c + = c c. 9

20 The costt-vlue codes for 3 re: (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) } A (,,,,, ) } A 7 (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) } A (,,,,, ) } A 8 (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) } A (,,,,, ) } A 9 (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) } A 3 (,,,,, ) } A (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) } A 4 (,,,,, ) } A (,,,,, ) (,,,,, ) (,,,,, )

21 (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) } A 5 (,,,,, ) } A (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) (,,,,, ) } A 6 (,,,,, ) (,,,,, ) Sce both d re geertors of trstvely o the set { 4}, ccordg to Theore 6. A * GF (3), the trsfortos d ct For we hve I {4,5,6}, d so S. Alyg (A ) Ab, wth b ( ) (cf. Theore 5) yelds the ch of trsfortos: A A9 A A A5 A6 A8 A A7 A A3 A A wheres ( A 4 ) A4, llustrtg Theore 6. For we fd I {,,3 } d S 6, gvg rse to (A ) Ab, wth b (6 ). Now we get the orbts A 4 A4 d A A A A A5 A A8 A9 A7 A A3 A6 A (v) For 7 we hve L, d K. I ths cse ether or s geertor of GF (7) *. For, we hve I {,,3,4 }, I {5,6,7,8}, d so S, S 9. The trsforto (A ) Ab wth b ( ) od 7 roduces the orbts: A A, A3 A4 A9 A A6 A5 A A A A, 4 A A3 A A5 A7 A6 A8 A4 ',. whle (A ) A b wth b (9 ) od 7 gves A A, A A6 A4 A A3 A5 A9 A A A, 4 A7 A3 A8 A5 A A6 A A4

22 8 8 Ths shows tht I d tht I, d hece tht ether or re geertors of G A, whch s ccordce wth Theore 6 (). I order to geerte G by sgle erutto we c tke 5, sce 5 ers A to be geertor of GF (7) *. Ideed 5 (A ) Ab wth b 5( ) od 7 yelds, strtg wth A, successvely A -sets wth dces, 3,,, 5, 4, 6, 8,, 6, 9, 7, 4, 5, 3,,. Ths shows tht the order of 5 s equl to 6, d lso tht 5 cts trstvely o the fly of sets { } ( cf. Theore 6 () d ()). A 6. Costructg A fro A I ths secto we shll dscuss ethod to trsfor vector A to vector b. Our ethod s bsed o the followg. We ssue tht the trx P, A GF( )*, (5) corresods to oe ( ) / -cycle whch we deote by d : ( d( ) d..... d ), d I, (33) ( ) / (cf. lso the roof of Theore 3). Ths s the cse ether f ord ( ) or f ord ( ) ( ) / d ( ) / s odd, ccordg to Theore. Corresodg to (33) we defe bry vector of legth ( ) /, such tht ts -th cooet s equl to the rty of the uber of d j, j, whch re I. Slrly, we troduce vector, the -th cooet of whch equls the rty of the uber of d, j, whch re I. It wll be cler tht the followg relto holds j = + (,,,, ). (34) Now, let be bry vector of legth ( ) / reresetg soe rtto, d let vl () =. Fro (33), fro the defto of P (5) d fro the defto of the cooets of, we y coclude tht d d( ) So, sce d, we c wrte od, ( ) /. (35) vl () = ( ) / ( ) / d d ( ) / ( ) d. (36)

23 We defe trslto vector s s follows. If d j j, j k, d d k soe k, k ( ) /, we defe (s ), j k, wheres ll other cooets re ut equl to zero. So, the bove teger k s the sllest dex wth d k k. Forlly, we c obt s by d j s = (,...,,,..., ) Q, (37) where the vector t the rhs cots k oes followed by ( ) / k zeros, whle the erutto trx Q hs eleets q, j f j d d q, j otherwse. Fro ow o, we shll ly the vectors d oly wth resect to. We lso troduce the secl vectors Q, c = ( + )Q = c Q. (38) THEOREM 7 Let ord () be equl to, or to ( ) / wth ( ) / odd. () f A, the b = + s A, uless = c ; () for ord () =, the trslto () gves oe-to-oe g A A, GF( ) \{, }, A \ { c } A d A A \ { }; () for ord ( ) ( ) / d beg odd, the trslto () gves oe-to-oe g A A, GF() \{, }, A A \ { } d A \ { c } A. Proof. Fro the codtos of Theore 7 d by lyg Theore, t follows tht the trx P, defed (5), reresets cyclc erutto of legth ( ) /. We oly hve to tke to ccout the chge the cotrbuto to vl () due to the cooets,...,. These cotrbute to vl (), lyg (36), out of k d dk ( ) od, where the sgs re detered by the cooets of. For d k, we oly hve ( ) = for those ostos where, for k, due to the defto of k. But these re recsely the ostos Hece, we fd vl (b) vl () = For d k k we hve d k k ( ) k d k, where b hs oes. k k for ( b ). (39) d b d for d k k =, b d so the secod ter the rhs of (39) lwys equls k. We coclude tht vl (b) vl () = k k ( ). The oly exceto occurs whe d j for j ( ) /. k d k j d k 3

24 I tht cse k s ot defed, d = c = c Q. Sce ( c ) d c wrte ( c ) d = vl ( c ) =, d hece by lyg (36) we fd ( ) / ( ) ( ), vl ( ) = So, cse tht ord ( ), whe we hve ( ) / ( ) / ( )/ ( ) / (, ( ) /, we ). (4), t follows tht vl ( ) vl ( c ) =, (4) O the other hd, we lso hve (cf. (8)) vl ( ) + vl ( c ) = ( ) / Reltos (4) d (4) together yeld, usg L, L. (4) vl ( ), vl ( c ) =. (43) I cse tht ord ( ) ( ) /, whe ( )/, we obt sted of (4), the equlty vl ( c ) vl ( ) =, (44) resultg vl ( c ) = vl ( ). (45) Stteets () d () ow follow edtely fro (43) d (45). The roerty of the gs () d () beg relly oe-to-oe, follows fro the fct tht the trsfortos c esly be reversed. Isted of the vector s s defed (37), we the hve vector s,defed by teger k, such tht k s the sllest dex wth d k k. As for the codto for ord (), the followg le y be useful. LEMMA 4 () If ord (), the 3 od 8. () If ord () ( ) /, the od 8. Proof. () Let ord (). Ths les ( )/ would be squre, sy, d hece whch cotrdcts GF( ) d. So, ths cse we hve 3 od 8.. Assue od 8, the for 4

25 () I geerl oe hs tht the squres GF( )* costtute subgrou of dex. Fro the codtto () we hve tht <> s subgrou of dex. Sce y grou hs oly oe subgrou of dex, t follows tht s squre, d cosequetly od 8. REMARK We rerk tht the lctos Le 4 c ot be reversed. For exle, ord () 4 d ord 43 3 () 5. we lso ehsze tht ord () ( ) / does ot ly tht ord () s odd. A couterexle s 7 wth ord 7 () 8. For vlues of other th, we were ot ble yet to derve results slr to those reseted Theore 7. Let us cosder the cse =, d ssue tht geertes GF ()*. I ths cse the erutto (33) s detcl to the oe for. Sce the sets I d I wth resect to re equl to the sets I d I wth resect to, the vectors d for re equl to d for. Exresso (35) ow yelds vl () = ( ) / ( ) ' ( ), (46) d sted of (38) we ow c wrte for the chge vlue due to the trsforto b = + s, the exresso vl (b) vl () = = k ( ) ( ) ( k ) k 3 ( ) ' k k = ( b k k )( ) k = (( ) ). (47) 3 We see tht for = the chge vlue s geerl ot the se for the vrous vectors of set A, but deeds o k. The se s true for other -vlues. Exles For 7 we hve ord 7 () 3 (7 ) /. Sce the order of s odd, we y ly Theore 7. For, I { }, I {,3}, d ( 3) d = (,,). For = (,,) A 5, we fd k, sce d. Sce Q I ths sle cse, t follows tht s = (,,) Q (,, ). So, b = + s = (,,), whch deed s eleet of A 6. By slr costructo we = (,,) A 3 oto b = (,,) A 4. The secod vector of A 3,.e. (,,), hs o ge uder ths g, llustrtg Theore 7 (), sce (,,) = c = c, d 3. O the other hd, the vector (,,) = s ot the ge of vector A. 5

26 For, we hve ord (), or equvletly, geertes GF () * (cf. Secto 5 ()). For ths -vlue, we hve I {, } d I {3,4,5}. The 5-cycle of (33) s equl to d ( 4 3 5), d hece = (,,,, ). The trx Q hs the for Q =. For = (,,,, ) A, we fd k 3, sce d, d d d 4 3. It follows tht s = (,,,, ) Q = (,,,, ). So, b = + s = 3 (,,,,), whch deed s vector A. If we tke = (,,,,) A, the k s ot defed, llustrtg Theore 7 (), sce c = c Q = (,,,, ) Q = (,,,,) A d. Tkg for the vectors (,,,, ) d (,,,, ), both fro A, yelds vectors b equl to (,,,, ) d (,,,, ), resectvely, whch re A 3. The thrd vector (,,,,) of A 3 s the coleet of c, so equls thus cofrg g Theore 7 (). For 3, we hve tht both d re geertors of GF (3) (cf. Secto 5 ()). Frst we tke, for whch we hve I {,,3 } d I {4,5,6}. The 6-cycle of (33) s equl to d = ( ), d hece = (,,,,,). The trx Q corresodg to d hs the for Q. We derve c = c Q = (,,,,,). Now, vl( c ) = 6 = 3 od 3, d so c A3 A. Accordg to Theore 7 (), ths vector hs o ge A 4 = A. Furtherore, vl( ) = 5 od 3. So, s A 5. Ag ccordg to Theore 7 (), ths vector s ot the ge of vector A 4. The four vectors A A 4 re ed oto the four other vectors of A 5 the followg wy: (,,,,,) (,,,,,), (,,,,,) (,,,,,), 6

27 (,,,,,) (,,,,,), (,,,,,) (,,,,,). I order to llustrte the rerk rght fter Theore 7, we tke. Of course, we hve the se 6-cycle ( ), fro whch we fer tht I {4,5,6}, I {,,3 }, d hece = (,,,,,), whch c lso be derved fro by lyg (34), d cosequetly = Q = (,,,,,). The vectors of A 3 re ed the followg wy. = (,,,,,) wth k s ed oto b = (,,,,,) A 4, = (,,,,,) wth k s ed oto b = (,,,,,) A, = (,,,,,) wth k 5 s ed oto b = (,,,,,) A, = (,,,,,) wth k s ed oto b = (,,,,,) A 4, = (,,,,,) wth k s ed oto b = (,,,,,) A. k These results coly wth (47), sce the vlue of (( ) ) / 3 s equl to, 3, d 8 for k, d 5, resectvely. The erutto d of the tegers of the set I {,,...,( )/ } (33) s wrtte such wy tht d. If we choose for dfferet teger t the frst osto, sy d t, oe c esly verfy by djustg the roof of Theore 7, tht oe obts oe-to-oe g fro A to A wth slr roertes. t Exle Tke 3 d d ( ). The corresodg rty vector equls = (,,,,,), d the corresodg trx Q s Q =. Ths yelds the followg trsfortos for the vectors of A 3 : 7

28 = (,,,,,), d, k, s (,,,,, ) b = (,,,,,);, = (,,,,,), d, d 4, k, s (,,,,, ) b = (,,,,,); = (,,,,,), d, k, s (,,,,, ) b = (,,,,,);,, = (,,,,,), d, d 4, k, s = (,,,,,), b = (,,,,,); = (,,,,,), d, d 4, d 5 3, b = (,,,,,). d 3 4, k 6, s (,,,,, ), d, d 6, As oe c see, ll b-vectors re A 5, whch cofrs the rerk o the revous ge wth t. 7. Deterg the sg of the Guss su G () 5 I ths secto we shll derve the correct sg(s) for the Guss su G () (cf. (9)). Frstly, we reset d rove theore cocerg the tegers whch were troduced Secto. THEOREM 8 () Ne N o A / for y odd re ; () f od 8, the No () f 3 od 8, the (v) f od 8, the (v) f 3 od 8, the No No No A / d A / d Ne Ne A /- d A / d Ne Ne 3 6 N e d A /; A /+; A /; A /+. N o, {,,..., } Proof. () These equltes re edte cosequece of the equltes (cf. Secto ) ( )/ c d c ( ) ( Ne N o ) for. () d () Assue od 4. The A / s eve s follows fro () d Theore 6 (v). Let the words (vectors) A,, be ordered soe wy. So, 8

29 we hve ordered blocks wth ( ) / colus d N( ) rows for, d N( ) rows for. Chgg colu j zeros to oes, d vce vers, yelds oe-tooe corresodece betwee the words A wth zero o osto j d the words A wth oe o tht osto for ll GF( ) d j ( ) /. Sce A / s j eve, the th j colu of A hs eve uber of zeros. It follows tht the th j colu of A kj wth k hs eve rty f d oly f k s odd.. We coclude tht A cots odd uber of colus wth odd rty, d hece lso odd uber of rows wth odd rty, f d oly f the equto kj, k eve, j {,,...,( ) / }, hs odd uber of solutos or, equvletly, f d oly f the equto lj, l, j {,,...,( ) / } hs odd uber of solutos GF( ). Becuse of the syetry betwee l d j, ths lst codto s stsfed f d oly f s squre GF( ). Sce ths s true f d oly f od 8, we hve ths cse tht A cots odd uber of rows wth odd rty for od 8 d eve uber of such rows for 3 od 8. Fro Theore 6 t follows tht for od 8 A = A /+( A / ), d for 3 A =( A /+)+ A /. Sce both cses A / s eve, the reltos () d () ow follow. (v) d (v) Assue tht od 4, d so A / s odd. I slr wy s the revous cse we derve tht A cots odd uber of rows wth odd rty f d oly f the equto kj, k odd, j {,,...,( ) / }, hs odd uber of solutos GF( ). Equvlet to ths codto s tht the equto lj, l, j {,,...,( ) / } hs odd uber of solutos. Sce od 4, ths s true f d oly f 3 od 8. Hece, we hve ths cse tht A hs odd uber of rows wth odd rty f 3 od 8 d eve uber of such rows f od 8. The reltos (v) d (v) ow follow s the revous cse. As kd of corollry we ow c stte the colete exressos for the Guss su G (). THEOREM 9 (Guss 85) For y odd re oe hs G() f od 4, d G() f od 4 Proof. Fro Theore 8 t follows tht N N for y odd re. Substtutg e o ths result the relto gves the bove result. ( ) ( )/ ( Ne No ) (cf. the ed of Secto ), 9

30 8. Orderg the vectors of ( )/ GF () two dfferet wys I order to del wth cses where Theore 7 cot be led,.e. whe ord () s less th ( ) /, or whe ord () = ( ) / d eve, we wll use the followg le. Ths le wll hel us to cosder the trsforto of Theore 7 slghtly dfferet cotext d to geerlze tht trsforto. I the le d the reg rt of ths reort we shll wrte bry ubers reversed order,.e. fro rght to left. Accordgly, bry ddto,.e. wth crry d deoted by, wll be crred out fro left to rght. The reso to hdle the bry reresetto of ubers ths wy, s to djust to the cycle otto of eruttos where the eruted objects re ordered fro left to rght. LEMMA 5 () Let u be soe fxed bry vector of legth. For y v GF ( ) we defe trsforto v w = v + s, where s s defed such tht s for k, where k s the frst dex vlue wth vk u k, whle s for k. Whe reetedly lyg ths trsforto, strtg wth the vector v := u, oe obts lst v, v = v + s, v = v + s, of ll vectors of GF ( ), edg wth the vector v = u c ; () If oe lbels the vectors ths lst fro utl, the e-th vector v e c be obted drectly fro v u by v e = u e, where e stds for the bry reresetto of legth of the teger e ( reversed order); () If v l s the l-th vector the lst, the v l e +u =(v l u) e for l e. Proof. The stteet the Le s trvl for u =. I tht cse the order of the vectors the lst s recsely the turl order of the tegers,,.,, whe the vectors re terreted s the bry reresettos of these tegers wrtte fro rght to left. Furtherore, t c esly be see tht lst strtg wth rbtrry u c be obted by ddg ( GF ( ) ) the vector u to ll vectors of the lst corresodg to u =. The lst s ot cyclc d ctully eds wth u c, sce for tht vector the trsforto s ot defed. The reg rt of the Le follows esly. Exle For 3 d u =, we obt the followg lst by reetedly lyg w = v + s: v s 3

31 The word v 6 c be obted fro v (=u) = by v =. Notce, tht whe lyg Le 5 uder the codtos of Theore 7, wth ( ) /, v = Q, A d u : = the rty vector corresodg to erutto d, the result s w = bq, b A (cf. lso (37) d (38)). Oe could sy tht Q s the vector d-reresetto, d therefore we wrte d := Q, d d slrly b : bq. Slrly, the vector s Q (37) s lso d-reresetto. Usg ths otto d cobg Theore 7 d Le 4 wth u =, rovdes us wth the followg theore. I ths theore d stds for the erutto (33), whle deotes the ddto of two bry ubers of legth ( ) / THEOREM Let ord () be equl to, or to ( ) / wth ( ) / odd, d let A be rereseted by d d-reresetto. Let furtherore e be the bry reresetto of legth ( ) / of soe ostve teger e d wrtte reversed order. The the vector b d, defed by b d + = ( d + ) e s the d-reresettve of vector b A e. Exle Let 3. The s geertor of GF (3) *. So, the trx P reresets the erutto d ( ) whch defes the rty vector = (,,,,, ). We strt wth the vector = (,,,,, ) A or wth d = Q = (,,,,, ) wth Q s defed Secto 6. Frst we ly Theore wth e rereseted by the bry vector e = (,,,,, ). Successvely, we fd d = (,,,,, ), ( d + ) e = (,,,,, ), b d = (,,,,, ) d flly b = b d Q = (,,,,, ) A. Actully, ths costructo s othg else s the costructo of Theore 7, forulted slghtly dfferet cotext. Next, we tke e 7, rereseted by e = (,,,,, ). Strtg wth the vector = (,,,,, ) A, we costruct successvely d = (,,,,, ), d + = (,,,,, ), ( d + ) e = (,,,,, ), b d = (,,,,, ) d b = b d Q = (,,,,, ), whch deed s vector A 8. I cse tht we wrte the erutto d dfferet wy, the erutto trx Q d the rty vector lso chge. Sy, we wrte d = ( ) wth corresodg rty vector = (,,,,, ). Now, f we strt wth = (,,,,, ) A 3 d wth e =, 3

32 we fd successvely d = Q = (,,,,, ), d = (,,,,, ), b d = ( d + ) e = (,,,,, ), b = b d Q = (,,,,, ) A 8. Exle As other exle we tke the cse d e = (,,,, ). We kow lredy tht, d = ( 4 3 5), = (,,,, ), whle Q hs the for Q. We strt wth the vector = Q =, whch s A 3 levg out brckets d co s, d = =, d A. Subsequetly, we fd, + =, b d + = ( d +) e =, b d =, d flly b = b d Q = A 4. 5 I slr wy we coute lst cotg ll 3 vectors of GF (). Below we reset the colete lst. d + b d = A 3 A 4 A 5 A 6 A 7 A 8 A 9 A A A A A 3 A 4 A 5 A 6 A 7 3

33 A 8 A 9 A A A A A 3 A 4 A 5 A 6 A 7 A 8 A 9 A A c = A As oe c see, the left lst strts wth the vector = (,,,, ) A 3 = A d eds wth c = (,,,, ) A = A, d cots ll 3 bry vectors of legth 5. The lst the ddle cots the corresodg vectors d +. Oe c verfy tht ths lst cossts g of ll bry vectors of legth 5, such tht the reversed vectors re turl order, strtg wth (,,,, ) d edg wth (,,,, ), thus corrobortg Le 5 d Theores 7() d 8. Exle For 7 we hve ord () = ( )/ = 3 whch s odd. The erutto trx P reresets the erutto d = ( 3). So, Q I. Furtherore, I {}, I {, 3}, =(,, ), 3, A = for 3, d A3 =. It follows tht Q = (,, ). I ths cse we hve the followg lsts: d + b d A 3 A 4 A 5 33

34 A 6 A A A c = A 3 Now, the left lst strts wth A d eds wth c A. Fro the bove exles t wll be cler tht the esest wy to obt lst of ll vectors ( ) / of GF ( ), ordered ccordg to cresg vlue (7), s s follows. We shll cll tht lst L d deote ts vectors (words) by,,., ( )/ the ext theore. THEOREM Let L be the lst of the bry vectors of GF() / turl order d wrtte fro rght to left. Trsforg ech v of ths lst to ( v + )Q yelds lst L whch s ordered such tht the vlue of the corresodg rttos creses by od whe gog to the ext word. The lst L strts wth Q d eds wth ( )/ = c. For c ord () oe hs A d A, wheres for ord ( ) ( ) / d odd, both d c re A. THEOREM Let C : { j : j {,,,...} }, be the set of vectors wth dces,,,.d of legth ( ) / the lst L, for y wth. () If ord (), the C = A for, whle C = A. () If ord () ( ) / wth ( ) / odd, the C = A for. Exle Tke 3. The =, 4, d the trx Q hs the for Q. Alyg Theore wth, yelds resectvely 34

35 Q =, 3 Q =, 6 = Q =, 39 = Q =, 5 = Q =. These re recsely the vectors of the costt-vlue code A 5. Observe tht the rocedure of costructg vectors L fro those L utotclly hlts whe we try to costruct 65, sce 65 s teger whch hs o bry reresetto of legth 6. By lyg Theore wth, we obt = Q =, 5 = Q =, 38 = Q =, 5 = Q =. These four vectors re the vectors of the fly A A 4. Here lso, the rocedure stos utotclly, sce 64 cot be rereseted s bry vector of legth 6. Flly, we gve the colete lsts L d L for our rug exle of 3. L L A 5 A 6 A 7 3 A 8 4 A 9 5 A 6 A 7 A 8 A 9 A A A 3 A 4 35

36 3 A 5 4 A 6 5 A 7 6 A 8 7 A 9 8 A 9 A A A A 3 A 4 A 3 5 A 4 6 A 5 7 A 6 8 A 7 9 A 8 3 A 9 3 A 3 A 33 A 34 A 35 A 36 A 37 A 3 38 A 4 39 A 5 4 A 6 4 A 7 4 A 8 43 A 9 44 A 45 A 46 A 47 A 48 A 36

37 49 A 5 A 3 5 A 4 5 A 5 53 A 6 54 A 7 55 A 8 56 A 9 57 A 58 A 59 A 6 A 6 A 6 A 63 A 3 9. The Morse-Thue sequece d relted roertes I order to del wth the rty of the weght of bry ubers, we troduce the wellkow Morse-Thue sequece M,,...,,..., (48) where stds for the rty of the uber of oes the bry reresetto of,.e. : f the weght s eve d : f ths weght s odd. Oe c esly derve the followg recursve rule, whch c serve s defto of M k k k c M ( ) M ( ) M ( ), () M, (49) where M() j deotes the subsequece of the frst j eleets of M. I the ext, subsequece of M shll lwys e subsequece of cosecutve eleets of M, uless stted otherwse. The followg sle roertes, collected le, c esly be rove by lyg (49). LEMMA 6 () M does ot cot subsequeces cosstg of three cosecutve detcl eleets. () The substtuto for ll, trsfors M to tself for ll k. k 37

38 () The substtuto rules,, together wth the ttor M (), geerte the sequece M. (v) For ll oe hs,. (v) For ll eve oe hs, d for ll odd,. Now, let M ',,..., k d M '',,..., k be subsequeces of M of legth k d of legth k, resectvely, cosstg of cosecutve eleets, d let furtherore be the uber of oes d be the uber of zeros ether of these sequeces. The we c derve the followg le. LEMMA 7 () I M oe hs for eve d y k, f, d f. k () I M oe hs for odd d y k, f, d f. () I M oe hs for y k,, excet f s odd d or k, whe d, resectvely. k k Proof. We shll rove the Le by ducto o k. Both stteets re trvlly true for k. Assue the stteets re true for soe k. Let M '' M ', k, k be subsequece of legth k 3. () If s eve, the k s eve. I cse tht k, we hve M'' M ',, or M'' M ',,. I the frst cse, M ' eds t d the secod cse t, becuse of Le 5 (v). Fro the ducto ssuto t ow follows tht for M '' the dfferece s equl to the frst cse, d to the secod cse. If, we hve M'' M ',, or M'' M ',, wth M ' edg t d k resectvely. Hece, g lyg the ssuto codto, the dfferece for M '' s equl to the frst cse, d to the secod. So, stteet () holds for k. () If s odd, the k s odd. Ag by Le (v), we y coclude tht M'' M ',, or M'' M ',,. Fro the ducto ssuto t follows tht cse tht f, the dfferece equls. If, ths dfferece equls. So, stteet () lso holds for k. By the rcle of thetcl ducto both stteets hold for ll k. Prt () s cosequece of () d (). I the ext we cosder the tegers the set {,,..., } wth ( )/ d wrtte bry, whch rovdes us wth ll bry ubers of legth ( ) /. As for our otto, whe wrtg od we e the teger {,,..., } whch s equl to 38

39 odulo. Ech of these tegers c be wrtte s j wth d j N( ), excet for N( ) the cse od 8 (cf. ()). For gve teger, oe obts d j by dvdg by. The quotet s equl to j d the reder to. Next we order ll tegers of the set {,,..., } lexcogrhclly wth resect to d j, d we cll the resultg lst L. It s obvous tht L c be wrtte s cocteto of sublsts L = B, B,..., B. (5) It wll lso be cler tht the set C (cf. Theore ) cossts of the bry reresettos (wrtte fro left to rght) of the tegers B,. For 3 od 8 the sublsts theselves re of the for For od 8 we hve B,,..., ( N( ) ),, (5) B,,...,( N( ) ). (5) B,,,..., ( N( ) ),, (53) B,,,...,( N( ) ), N( ). (54) THEOREM 3 Let be odd re such tht ord (). Let F be g defed by F( b) b od N( ), for ll b {,,..., }. () F s oe-to-oe g o {,,..., }. () F geertes orbts of legth k, wth k : ord (), o the set of eleets ot B d ech orbt cots recsely oe eleet of B, for y. () O the fly of sets { B } F geertes oe orbt of legth. (v) F duces erutto of the eleets of B, d the uber j B les orbt of legth k j, whch s equl to the order of od N( ) / ( j, N( )), for j N( ).. ( )/ (v) If bl bl... b, wth l: ( ) /, s the bry reresetto of b GF (), c the bl... bb l s the bry reresetto of Fb. () (v) If e deotes the uber of bry words B wth eve uber of oes d o the uber wth odd uber of oes, the. e o 39

40 (v) If F k s the g defed o {,,..., } by Fk ( b) b k od N( ), for y fxed k d, the F k duces erutto of the eleets of B f, d of the eleets of B f. The teger j B, j N( ), les orbt of legth k j whch s equl to the order of od N( ) / ( j k, N( )), f j k od N( ). The teger of F k. k B s fxed ot Proof. We rerk tht becuse of the ssuto ord (), we hve fro Le 4() tht 3 od 8. Therefore, we hve to tke (5) d (5) s defto of the sets B. () Fro F( b) F( c ) t follows tht b c od N( ), d so b c, sce both re less th N( ). Ths les the stteet. () Tke b B,. The F ( b) b od N( ). Puttg F ( b) b yelds the equto ( )( b ) od N( ). Sce b B, we hve b od. Hece, od, d so ord (). () Ths s cosequece of (). (v) Cosder b : j B, j N( ), The F( b) j od N( ). If j N( ), we hve by defto tht j B. If j N( ), we subtrct N( ) d obt ( j N( )) whch s clerly lso B. Alyg F,, o j B, j N( ), d uttg the result equl to j tself, yelds the equto ( ) j od N( ), or equvletly ( ) j od N( ). The result follows edtely. (v) If b l, the vlue of bl... b s b whch s equl to Fb, () sce b ( ) /, d hece b N( ). If b l, the vlue of bl... b s equl to b b N( ). Now, b / d so b N( ). Accordg to the defto of F, we g c detfy ths result wth Fb. () (v) Ths follows edtely fro (v) d (v), sce y te F s led o eleet of B, the rty of ts bry reresetto chges. (v) Cosder b : j B, j N( ). The Fk ( b) ( j k) od N( ). Slry s the roof of (v), t follows tht ths teger s lwys B. The vlue of the orbt legth k j s derved slr wy s the roof of (v). Alyg F k,, o j B, d uttg the result equl to j, yelds the equto ( )( j k ) od N( ). For j k od N( ), the result ow follows edtely. For j k we obt tht Fk ( k) ( k) k k. THEOREM 4 4

41 Let be odd re such tht ord () s equl to ( ) / d odd. Let furtherore G be the trsforto defed by G( b) b od N( ), for ll b {,,..., }. () G s oe-to-oe g o {,,..., }. () G geertes orbts of legth k : ord () ( ) / o the set of eleets ot B d ech orbt cots t ost oe eleet of B,. () O the fly of sets { B } G geertes two orbts of legth ( ) /. (v) G duces erutto of the eleets of \{ } j B les orbt of legth k j whch s equl to the order of od N( ) / ( j, N( )), for j N( ), whle s fxed ot. ( )/ (v) If bl bl... b, wth l: ( ) /, s the bry reresetto of b GF (), the bl... bb l s the bry reresetto of Gb ( ). (v) If e d o deote the uber of bry words B of eve rty d of odd rty resectvely, the. e o (v) If G s the g defed o {,,..., } by G ( b) b k od k N( ), for y fxed k d,, the G k duces erutto of the eleets of B f, d of the eleets of B \{ } f. The teger j B, j N( ), les orbt of legth k j whch s equl to the order of od N( ) / ( j k, N( )), f j k od N( ). The teger G k. k k B s fxed ot of Proof. Becuse of the ssuto ord () ( ) /, we hve fro Le 4 () tht od 8, d hece we hve to tke (53) d (54) s defto for the sets B. Prts () (v) re roved slr wy s the corresodg rts of Theore. As for (v), we rerk tht for j N( ), j d ( N( ) j) re both B, d tht the su of these two tegers s equl to N( ). Sce the bry reresetto of s equl to the ll-oe word... of legth ( ) / d sce ths uber s ssued to be odd, the set B cots s y words of eve rty s of odd rty. I order to rove (v), we frst rerk tht G ( j) ( j k) B for ll. Alyg k G k,, o the teger j B d uttg the result equl to ths teger, yelds the equto ( )( j k ) od N( ), lke the roof of Theore 3 (v), d the result follows slrly. Exle Frst we tke 3. So, N (3) 5, 6 N (3).3 d ord 3 (). Strtg wth b d lyg F reetedly rovdes us wth the followg orbt of legth : 4

42 F ( b), F 4 ( b) 5, F 8 ( b) 6 8 4, F ( b), F 5 ( b) 3 5, F 9 ( b) , F ( b) 3 3, F 6 ( b) 63 4, F ( b) , F 3 ( b) 7 7, F 7 ( b) 6 4. F ( b) 3 6, F ( b) b. I slr wy we obt the other four orbts of eleets ot B. Below we reset ll fve orbts bry reresetto: The frst eleet the vrous colus s chose such tht ech bry teger dffers recsely oe bt fro the teger left fro t the se row. By Theore 3 (v), the se holds ll rows. Oe could sy tht eghbourg colus re rllel t dstce. The trsforto F geertes the followg orbt o the fly { B } : B B B3 B7 B B5 B B B8 B4 B9 B6 B The words B re As oe c verfy, the trsforto F erutes these words ccordg to 4

43 Rerk tht the uercl vlues of b d Fb, () whch le the set {,,...,63}, re such tht F( b) b od 65 (65 = N( ), (cf. the roof of Theore 3). Equvletly, the bove seres of trsfortos c lso be dcted by where the coeffcet 3 s obted by.4 = 3 od 5 ( =N(3)). The sze 4 of the oly orbt B s greeet wth Theore 3 (v), sce N (3) 5 d ( j,5), j 4. I order to llustrte Theore 3 (v), we cosder the cto of the oertors F F, k {,,,3, 4}, o the set B {,3, 6,39,5}. F geertes the orbt d hs fxed ot ; F geertes the orbt d hs fxed ot 5; F geertes the or bt d hs fxed ot 39; F geertes the orbt d hs fxed ot 6; 3 F geertes the orbt d hs fxed ot 3. 4 Fro the bry reresettos, 3, 6, 39 d 5, t follows tht the orbt of F hs the roerty tht ech eleet hs edte successor wth ooste rty. The other four orbts of legth 4 do ot shre tht roerty. Slrly, there exsts uque oertor F for the set B wth the se roerty, e.g.: 4 3 F geertes the orbt d hs fxed ot 4; F geertes the orbt d hs fxed ot ; F geertes the orbt d hs fxed ot 6; 3 4 F geertes the orbt d hs fxed ot , k k k I the frst rt of the bove exle we observed tht the fve orbts of the oertor F ctg o the eleets ot B c be ordered such tht y word oe orbt s t dstce fro the corresodg word other orbt. It turs out tht ths heoeo s rt of ore geerl roerty of the oertor F, whch we shll rove below. THEOREM 5 Let be re such tht 3 od 8. Let O d ctg o the eleets ot B O j be two orbts of the oertor F. If the tl words these orbts re chose such 43

44 tht ther Hg dstce s equl to d, d ( ) /, the y two corresodg words O d O re t Hg dstce d. j Proof If the left ost bts of the two words t Hg dstce d re both equl to or both equl to, the rght ost bts re both ut equl to or both to, resectvely, uder the cto of F, d so the dstce does ot chge. If the left ost bts re d, the rght ost bts re ut equl to d by F, d hece the Hg dstce s g d fter the cto of F. Exle Next we tke 9 9, wth N(9) 7, (9).9 d ord () 8. (= F, cf. Theore 3) geertes the followg orbt o the fly of sets B 8} : F There re three orbts B B = {8, 37,., 493} of legth 6, d 8, resectvely: N 9 B B B3 B7 B5 B B6 B3 B8 B B B B B B B B B B { { { { Here g, we c see tht the trsforto F s lso estblshed by 44

45 j j ', wth j' j od N(9) (= 7). For we fd orbt of sze k = 8 whch s equl to the order of wth resect to j 7/(7,) = 7. We fd k 8. Slrly, we fd for j 3 orbt of legth 6 d for j 9 orbt of legth. Slrly, we vestgte the orbts of the oertor whch leves the set B = {, 9, 38,., 494} vrt uder ts cto F ( b ) b od N( ). Ag we fd orbts of legth 8, 6 d, together wth orbt of legth (cf. Theore 3 (v)). The resectve orbts re: , d the fxed ot 494 = , , F Flly, we shll llustrte Theore 3 (v), by cosderg the cto of F.9 F8 the set B {,,58,...,495}. Accordg to tht Theore B s vrt uder ths cto. We fd the followg orbts: o , , , whle 495 = the (oly) fxed ot s. B Next we tke exle such tht ord () ( ) / d eve. Exle 8 Let 7. We hve N(7) 5, N(7).7 d ord () 8 7. We shll show tht the oertor F, whch ths cse ( od8) s lso defed by F( b) b od N( ), hs slr roertes s the oertor Theore 3 where we delt wth the cse 3 od 8. Strtg wth b d lyg F reetedly, yelds the orbt

46 Slrly, we get orbt Altogether we obt 3 orbts of legth 8 cotg ll eleets whch re ot B6 55( ), 6, 33, 5, 67, 84,, 8, 35, 5, 69, 86, 3,, 37, 54, whch s slr to Theore (). The trsforto F( b) b od 55 geertes the followg orbts o the fly B 6 }: { B B B3 B7 B5 B4 B B8 B B B B B B B B B B Sce ord () 8 7, ths result s slr to Theore 3 (). The trsforto geertes fve orbts B : F Observe tht lwys F( b) F( j ) j ', wth j' j od 5. As secl cses we drw the reder s tteto to F(5 ).5 5 od 5, whch gves the sze of for the lst orbt. The legth 4 of the orbts corresodg to 6, 5 d 8 re detered by k k k = 4, beg the order of wth resect to 5= 3 5 () 3 N(7) / ( j, N(7)) for j, 3 d 5. Furtherore, k5 whch s the order of wth resect to N(7) / (5, N(7)) 5/ 5 3. Exle Now we tke 3. Sce ord whch s odd, the codtos of Theore 4 re stsfed. Furtherore, we hve N( ) 3.89, d so N(3) 89. It ers tht, rt fro the fxed ots d, the oertor G geertes 8 orbts of legth o.e. the orbts cotg resectvely B 46

47 .3 =, 9.3 =, 9.3 =, 33.3 =, 3.3 =, 5.3 =,.3 =, 3.3 =. As oe c verfy, the words the frst four orbts ll hve eve rty d the words the secod four orbts ll hve odd rty. The eght o-trvl orbts ll hve the se legth ths cse, becuse N (3) 89 s re d so ( j,89) for ll relevt j (cf. Theore 4). Slrly, let oerte o eleets whch re ot. If we tke the teger B, G B we obt successvely eleets lyg the sets B, B, B4, B8. B6, B9, B8, B3, B3, B6, B d bck to. Strtg wth 5 B yelds successvely the sets B 5 B5, B, B, B7, B, B, B, B9, B5, B7, B4 d bck to B5. Cotug ths wy, we get orbts of legth cotg ll eleets ot. Next, we cosder the oertor G whch s defed by the g G ( ) b b. Theore 4 (v) sttes tht the set B s vrt uder ts cto. We fd for the orbts B, rt fro the fxed ot 88.3, 8 orbts of legth (observe tht ord N( ) = ord () ths cse, sce ( N( ), ) (89,3) ). The orbts re:, +.3, + 3.3, + 7.3, + 5.3, + 3.3, , , , , , wth rtes,,,,,,,,, d, resectvely. Slr orbts exst cotg + 9.3, + 9.3, , + 3.3, + 5.3, +.3 d B REMARK I the bove exle we hve the eculrty tht both the o-trvl orbts s the orbts of eleets ot B s well, both hve the se legth. Ths c lso occur other cses,.e. f ord () ( ) / d odd (the codtos of Theore 4), N( ) s re d s ot Mersee re. We c see ths the followg wy. Fro the ( )/ codtos for ord (), we kow tht od 8 d tht N( ). Let l l be the order of wth resect to N( ), the kn( ) for soe ostve teger k. l ( )/ It follows tht l ( ) /, d hece d k. So, ether k or k. But k les tht N( ) s Mersee re, whch s forbdde. We coclude tht k d l ( ) /. The ext theore rtly geerlzes Theore 3 (), (v) d Theore 4 (), (v), d uts soe of the results of those theores slghtly dfferet otto. B 47

48 It s obvous tht the gs F k d G k re ctully the se fucto wrtte dfferet otto. More recsely, defg the fucto Hk, :{,,..., } {,,..., }, Hk, ( b) b k od N( ) yelds s secl cses F H, d G H,. Aother secl cse s the g H e., THEOREM 6 Let be odd re. Let H e be the g defed by, N( ), for ll b {,,..., }. () H, d () H, d resectvely. F e, e, s oe-to-oe g o {,,..., }. F e, e, duces erutto of the eleets of Proof. () Ths s show the se wy s Theore 3 (). () Cosder the cto of H e, d tke : e b j B. The, e ( ) e e e e, ( ) e H e b b od, B d B, H b j j od N( ). If j N( ), we hve by defto tht the result s g B. If j N ( ), we subtrct N( ) d obt e ( j N( )) e whch clerly s lso B. For H e, the roof s slr. e We rerk tht uder the codtos of Theores 3 d 4, we derved the equlty e o for d, resectvely, but tht oe of the Theores -4 gves swer to the questo of the reltve szes of o d for other vlues of, d ether for cses whe the codtos Theores 3 d 4 re ot stsfed. By lyg the result of Guss (Theore 9), we c get rtl swer to tht questo s the ext theore wll show. We toduce the sybol, defed s f s eve, d f s odd. THEOREM 7 Let be odd re wth ord () or wth ord () ( ) / d odd. Let furtherore ' be the subsequece,,..., of the Morse-Thue sequece M ( N ( ) ) M, where. If e d o deote the uber of eleets M ' of resectvely eve d odd weght, the the followg reltos hold: () = for ll stsfyg, ; e o e 48