Necessary and sucient conditions for some two. Abstract. Further we show that the necessary conditions for the existence of an OD(44 s 1 s 2 )

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1 Neessry n suient onitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos, M. Mitrouli y, n Jennifer Seerry z Deite to Professor Anne Penfol Street Astrt We give new lgorithm whih llows us to onstrut new sets of sequenes with entries from the ommuting vriles 0 with zero utoorreltion funtion. We show tht for twelve ses if the esigns exist they nnot e onstrte using four irulnt mtries in the Goethls-Seiel rry. Further we show tht the neessry onitions for the existene of n OD(44 s 1 s 2 ) re suient exept possily for the following 7 ses: (7 32) (8 31) (9 30) (9 33) (11 30) (13 29) (15 26) whih oul not e foun euse of the lrge size of the serh spe for omplete serh. These ses remin open. In ll we n 398 ses, show 67 o not exist n estlish 12 ses nnot e onstrute using four irulnt mtries. We give new onstrution for OD(2n) nod(n + 1) from OD(n). The full OD(44 s 1 s 2 s 3 44 ; s 1 ; s 2 ; s 3 )given in this pper yiel t lest 68 equivlene lsses of Hmr mtries. Key wors n phrses: Autoorreltion, onstrution, sequene, orthogonl esign. AMS Sujet Clssition: Primry 05B15, 05B20, Seonry 62K05. 1 Introution Throughout this pper we will use the enition n nottion of Koukouvinos, Mitrouli, Seerry n Krels [2]. We note from [3] tht we hve totest 1 4 n2 = 484 ses. We n 398 ses, show 67o not exist n estlish 12 ses nnot e onstrute using four irulnt mtries. There re 7 open ses whih oul not e foun euse of the lrge size of the serh spe for omplete serh. 2 New orthogonl esigns Theorem 1 An OD(44 s 1 s 2 ) nnot exist for the following 2;tuples (s 1 s 2 ): Deprtment of Mthemtis, Ntionl Tehnil University ofathens, Zogrfou 15773, Athens, Greee. y Deprtment of Mthemtis, University ofathens, Pnepistemiopolis 15784, Athens, Greee. z Shool of IT n Computer Siene, University ofwollongong, Wollongong, NSW, 2522, Austrli. 1

2 (1 7) (1 15) (1 23) (1 28) (1 31) (1 39) (1 42) (2 14) (2 30) (3 5) (3 13) (3 20) (3 21) (3 29) (3 37) (3 40) (4 7) (4 15) (4 23) (4 28) (4 31) (4 39) (5 11) (5 12) (5 19) (5 27) (5 35) (6 10) (6 26) (7 9) (7 16) (7 17) (7 25) (7 28) (7 33) (7 36) (8 14) (8 30) (9 15) (9 23) (9 28) (9 31) (10 17) (10 22) (10 24) (11 13) (11 16) (11 20) (11 21) (11 29) (12 13) (12 15) (12 20) (12 21) (12 29) (13 19) (13 27) (14 18) (15 16) (15 17) (15 20) (15 25) (16 19) (16 23) (16 28) (17 23) (19 20) (19 21) Proof. These ses re eliminte y the numer theoreti neessry onitions given in [1] or[2, Lemm 3]. Exmple. To illustrte how we use the numer theoreti onitions to estlish the nonexistene of n OD(4n 11 20) we onsier the prout = now thisis numer of the form 4 (8 + 7) whih nnot e written s the sum of three squres n hene n OD(4n 11 20) nnot exist. Remrk. A omputer serh, whih we elieve ws exhustive, ws rrie out whih les us to elieve tht 1. there re no 4-NPAF(7 19) sequenes of length there re no 4-NPAF(3 31), 4-NPAF(5 30), 4-NPAF(6 29) n 4-NPAF(8 27) sequenes of length 9. This mens tht there re lso no 4-NPAF(1 5 30), 4- NPAF(1 6 29) n 4-NPAF(1 8 27) of length there re no 4-NPAF(2 41) sequenes of length 11. This mens tht there re lso no 4-NPAF(1 2 41) sequenes of length there re no 4-NPAF(6 37) sequenes of length 11. Lemm 1 OD( ) n n OD( ) o not exist (this is prove theoretilly). The Germit-Verner Theorem sys tht if n OD( ) exists then n OD( ) will exist, n if n OD( ) exists then n OD( ) will exist. Hene the OD( ) n OD( ) o not exist. Lemm 2 The following OD( ; ) n OD(44 43 ; ) nnot e onstrute using four irulnt mtries in the Goethls-Seiel rry: (6 37) (1 6 37) (10 33) ( ) (12 31) ( ) (13 30) ( ) (14 29) ( ) (16 27) ( ) (19 24) ( ) (20 23) ( ) Proof. By the Germit-Verner theorem if n orthogonl esign OD(n x 1 x 2 x u;1 x u ) with u i=1 x i = n ; 1 exists, n 0(mo 4), then n OD(n 1 x 1 x 2 x u;1 x u ) exists. Now for eh of the ses in this lemm we hve nod(44 43 ; ) n tht is y the Germit-Verner theorem n OD( ; ). Using the sum-ll mtrix metho we write 1 = , = n 43; = We require the sum-ll mtrix to e 34 orthogonl mtrix with the rst row ontining n 0 the seon row ontining n 4 in some orer n the thir row ontining n 4 in some orer. 2

3 However, s we illustrte for OD( ), this is not possile for the ses mentione in the enunition. Using the sum-ll mtrix metho for OD( ), 1 = , 20 = n 23 = (;1) There is no wy to form n orthogonl mtrix unless oth 20 n 23 n e written s the sum of 3 squres. 2 Theorem 2 Therere OD(44 s 1 s 2 s 3 44;s 1 ;s 2 ;s 3 ) onstrute using four sequenes to otin four irulnt mtries for use in the Goethls-Seiel rry for the following 2;tuples: Corollry 1 By suitly hoosing the vriles of the known OD(44 s 1 s 2 s 3 44 ; s 1 ; s 2 ; s 3 ) to e reple y1 these le to t lest 36 lgerilly inequivlent Hmr mtries of orer 44. Bysuitlyhoosing the vriles of the known OD(44 s 1 44 ; s 1 ) to e reple y 1 these le to t lest 12 more lgerilly inequivlent Hmr mtries of orer 44. Corollry 2 By suitly hoosing the vriles of the known OD(44 1 s 1 35 ; s 1 ) to e reple y 1 we otin t lest 20 lgerilly inequivlent skew-hmr mtries of orer 44. The numer epens on whether eh skew-hmr mtrix is equivlent to its trnspose or not. 3 New Algorithm The lgorithm previously use to n OD vi four sequenes of length t 10 ws prohiitively slow for length 11. Hene we trie new lgorithm, whih epene on the previous lgorithm, to n rst W (4t k) me with four sequenes of length t with PAF =0orNPAF = 0. In the new lgorithm MAT LAB ws use to set up series of equtions to e solve for eh iniviul k n then ll solutions to these equtions were foun. Exmple. We illustrte the lgorithm y trying to onstrut the OD( ). We rst notie tht 11 hs unique eomposition into squres 11 = , while 27 hs three eompositions into four squres. All three n e use in this onstrution s they must e le to e use in n integer mtrix (the sum-ll mtrix) whih is orthogonl. Hene we use27= = = Sowehve the mtries " ;1 5 # " ;1 4 ;1 3 # or " ;3 3 We now ll eh of the positions whih re represente y 0y one of 17 vriles x 1 x 2 x 17. Wenow use MATLAB to expn the rst rows to mke four irulnt mtries with row inner prout zero: this orrespons to forming four sequenes with PAF = 0. The equtions will e those tht involve somex j,1j 17 with, n those whih hve no terms in. This gives t most 6 inepenent equtions. A serh isnow me through the 17 vriles, llowing them to ssume the vlues 0 1, where six of them must lwys e zero, n using the extr onstrints tht 3 #

4 3X i=1 x i = ;1 5X i=4 x i = ;1 X11 i=6 x i =3 X17 i=12 x i =0: We strt with the following four sequenes of length 11 n PAF = We reple the 1 yvrile suh s n we reple the 17 zeros y the vriles. Thus we hve the sequenes ; ; x 1 x 2 x 3 ; ; ; ; ; x 4 x 5 x 6 x 7 x 8 x 9 ; ; x 10 x 11 ; x 12 x 13 x 14 x 15 x 16 x 17 We then use MATLAB to set up series of equtions, tht when solve, yiel, mong others, the following solution: x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x 15 x 16 x 17 0 ; 0 0 ; ; 0 ; ; 0 0 We now reple the vriles in the originl four sequenes y these solutions to otin the OD( ). 2 Remrk. Using this lgorithm we teste ll unknown two vrile ses n foun 7 ses whih we were unle to resolve ue to the extremely lrge serh spe. We estimte tht omplete serh for the OD( ) using this lgorithm requires 2 37 opertions. 2 4 New Results Theorem 3 Write X( ) = fe 1 x 1 e 2 x 2 e n;1 x n;1 e n x n g, Y ( ) = ff 1 y 1 f 2 y 2 f n;1 y n;1 f n y n g for the sequenes of length n, NPAF=0,where e i n f i re hosen from where, re ommuting vriles n x i, y i hve elements 0 1 n the sequenes X(1 1) n Y (1 1) hve NPAF = 0. Suppose ours totl of s 1 times n totl of s 2 times then we sy the two sequenes we hve re 2-NPAF(n s 1 s 2 ). Write i = if e i = n i = if e i = for i = 1,...,n, n similrly, i = if f i = n i = if f i = for i = 1,...,n. Then (i) X( ) Y ( ) n Y ( ) X (; ;) where Z enotes the reverse of the sequene Z or n fe 1 x 1 e 2 x 2 e n;1 x n;1 e n x n n y n n;1 y n;1 2 y 2 1 y 1 g ff 1 y 1 f 2 y 2 f n;1 y n;1 f n y n ; n x n ; n;1 x n;1 ; 2 x 2 ; 1 x 1 g re two sequenes with elements f0 g with NPAF = 0. These sequenes re 2-NPAF(2n 2s 1 2s 2 ). 4

5 n (ii) If x n;1 n y n;1 re oth zero then the sequenes fe 1 x 1 e 2 x 2 n y n e n x n n;2 y n;2 2 y 2 1 y 1 g ff 1 y 1 f 2 y 2 ; n x n f n y n ; n;2 x n;2 ; 2 x 2 ; 1 x 1 g re two sequenes with elements f0 g with NPAF = 0. These sequenes re 2-NPAF(2n ; 2 2s 1 2s 2 ). (iii) Similrly with 4-NPAF(n s 1 s 2 ), X( ), Y ( ), Z( ) n W ( ) we hve X( ) Y ( ) Y ( ) X (; ;) Z( ) W ( ) n W ( ) Z (; ;) where Z enotes the reverse of the sequene Z re 4-NPAF(2n 2s 1 2s 2 ). (iv) Similrly with 4-NPAF(n s 1 s 2 ),iftheseon lst element of eh of the four sequenes is zero thenproeeing s in (ii) we otin 4-NPAF(2n ; 2 2s 1 2s 2 ). (v) Similrly if there re 4-NPAF(n s 1 s 2 ),ntheseon lst element of two of the sequenes is zero n the lst element of two of the sequenes is zero then omining the methos of (ii) n (iii) we n get 4-NPAF(2n ; 2 2s 1 2s 2 ). Proof. The proof follows y writing out the sequenes n heking the NPAF. Exmple. We use to men ; n to men ;. To illustrte prt (v) of the theorem we note tht n re 4-NPAF(7 2 16) n 4-NPAF(7 4 16), respetively. In ft we note n re 4-NPAF( ) n 4-NPAF( ), respetively. We lso note tht n re 4-NPAF( ) n 4-NPAF( ), respetively n 5

6 re 4-NPAF( ) n 4-NPAF( ), respetively. Lemm 3 If there exist 2-NPAF(n s 1 s 2 ) then there exist 4-NPAF(n s 1 2s 2 ). Corollry 3 Sine there exist2-npaf(n s 1 s 2 ) for the vlues liste in the tle we get the orresponing lrger 4-NPAF(n s 1 2s 2 ). 2-NPAF(n s 1 s 2 ) ) 4-NPAF(n s 1 2s 2 ) (9 13) (10 2,2,26) (11 13) (12 2,2,26) (14 17) (15 2,2,34) (18 25) (19 2,2,50) (4 4,4) (5 2,2,8,8) (6 2,8) (7 2,2,4,16) (6 5,5) (7 2,2,10,10) (8 8,8) (9 2,2,16,16) (10 10,10) (11 2,2,20,20) (14 13,13) (15 2,2,26,26) Corollry 4 Using the previous theorem we see tht 4-NPAF(n s 1 s 2 ) ) 4-NPAF(2n 2s 1 2s 2 ) NPAF(5 1,18) NPAF(5 1,19) NPAF(5 2,17) NPAF(5 2,18) NPAF(5 3,17) NPAF(7 3,18) NPAF(5 4,16) NPAF(7 4,17) NPAF(7 4,18) NPAF(5 5,14) NPAF(5 5,15) NPAF(7 5,16) NPAF(7 5,17) NPAF(7 5,18) NPAF(5 6,14) NPAF(7 6,16) NPAF(7 7,14) NPAF(7 7,15) NPAF(5 8,11) NPAF(5 8,12) NPAF(5 9,10) NPAF(5 9,11) NPAF(7 9,12) NPAF(10 2,36) NPAF(10 2,38) NPAF(10 4,34) NPAF(10 4,36) NPAF(10 6,34) NPAF(14 6,36) NPAF(10 8,32) NPAF(14 8,34) NPAF(14 8,36) NPAF(10 10,28) NPAF(10 10,30) NPAF(14 10,32) NPAF(14 10,34) NPAF(14 10,36) NPAF(10 12,28) NPAF(14 12,32) NPAF(14 14,28) NPAF(14 14,30) NPAF(10 16,22) NPAF(10 16,24) NPAF(10 18,20) NPAF(10 18,22) NPAF(14 18,24) Theorem 4 The sequenes given in the Appenies n e use to onstrut the pproprite esigns to estlish tht the neessry onitions for the existene ofnod(44 s 1 s 2 ) re suient, exept possily for the following 12 ses whih nnot e onstrute from four irulnt mtries: 6

7 (5 38) (6 37) (8 35) (10 33) (12 31) (13 30) (14 29) (15 28) (16 27) (19 24) (20 23) (21 22): n the following 7 ses whih re uneie: (7 32) (8 31) (9 30) (9 33) (11 30) (13 29) (15 26) Remrk. There re 484 possile 2;tuples. Tle 1 lists the 398 whih orrespon to esigns whih exist in orer 44: 67 2-tuples orrespon to esigns eliminte y numer theory (NE). For 12 ses, if the esigns exist, they nnot e onstrute using irulnt mtries (Y). 7 ses remin uneie. P inites tht 4-PAF sequenes with length 11 exist n inites 4-NPAF sequenes with length n exist. 7

8 NE NE NE NE NE NE NE NE NE P NE NE NE 3 21 NE NE NE NE NE NE NE NE NE P NE NE 5 12 NE NE NE P 5 35 NE P 5 38 Y NE NE P P P Y NE NE 7 17 NE NE NE P NE 7 34 P 7 35 P 7 36 NE NE P P 8 30 NE P Y NE NE Tle 1: The existene of OD(44 s 1 s 2). 8

9 NE 9 29 P NE 9 32 P P 9 35 P NE P NE NE P P P Y NE NE NE NE P P P NE P P P NE NE NE NE NE P P NE P Y NE P P P P NE P Y P P P NE P P P P P Y P NE NE NE P P P NE P Y P NE NE P Y NE P P NE P P P P P P P P P P NE NE P P Y P P Y Y P Tle 1(Cont): The existene of OD(44 s 1 s 2). Referenes [1] A.V.Germit, n J.Seerry, Orthogonl esigns: Qurti forms n Hmr mtries, Mrel Dekker, New York-Bsel, [2] C.Koukouvinos, M.Mitrouli, J.Seerry, n P.Krels, On suient onitions for some orthogonl esigns n sequenes with zero utoorreltion funtion, Austrls. J. Comin., 13, (1996), [3] C.Koukouvinos n Jennifer Seerry, New orthogonl esigns n sequenes with two n three vriles in orer 28, Ars Comintori, (to pper). 9

10 A1 A2 A3 A4 Appenix A: Orer 40 (Sequenes with zero non-perioi utoorreltion funtion) Design ) (1 (1 4 32) ( ) (2 2 34) (2 4 32) ( ) ( ) ( ) (2 35) (3 31) (3 34) notinyet ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; 0 ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; 0 ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; 0 ; ; ; ; ; 0 ; 0 ; ; ; ; 0 ; 0 ; ; ; ; ; ; ; ; ; 0 ; 0 ; 0 10

11 A1 A2 A3 A4 Appenix A(ont): Orer 40 (Sequenes with zero non-perioi utoorreltion funtion) Design ) (4 ( ) ( ) ( ) ( ) (5 30) (5 33) (6 31) (7 31) ( ) ( ) notinyet ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; 0 ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; 0 ; ; ; ; 0 ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; 11

12 A1 A2 A3 A4 Appenix B: Orer 44 (Sequenes with zero non-perioi utoorreltion funtion) Design ) (1 (1 30) (1 34) (1 35) (1 37) (1 38) (1 40) (1 41) ( ) ( ) ( ) ( ) 0 ; ; 0 ; ; 0 ; 0 ; ; ; ; ; ; ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 0 ; ; ; 0 0 ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; 0 ; 0 ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; 0 ; ; 0 ; 0 ; ; ; ; ; 12

13 A1 A2 A3 A4 Appenix B(ont): Orer 44 (Sequenes with zero non-perioi utoorreltion funtion) Design ) (2 (2 37) (2 39) (3 35) (3 36) (3 38) (3 39) (3 41) (4 35) (5 36) (5 39) (7 37) ; ; ; 0 ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; 0 0 ; ; 0 0 ; 0 ; ; 0 ; 0 ; ; 0 0 ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; 0 0 ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 0 ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 13

14 A1 A2 A3 A4 Appenix C: Orer 44 (Sequenes with zero perioi utoorreltion funtion) Design 9 34) (1 ( ) ( ) ( ) ( ) (2 41) (4 37) (5 34) (5 37) (6 29) (6 33) (6 35) (7 30) ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; 0 0 ; ; ; ; 0 0 ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; 0 0 ; 0 ; 0 ; ; 0 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; 0 ; ; ; ; 0 ; ; 14

15 A1 A2 A3 A4 Appenix C(ont): Orer 44 (Sequenes with zero perioi utoorreltion funtion) Design 34) (7 (7 35) (8 27) (8 29) (8 33) (9 32) (10 31) (11 27) (11 28) (11 31) (12 25) (12 26) (12 30) ; ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; 0 0 ; ; ; 0 ; ; ; ; ; 0 ; ; 0 ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; 0 0 ; ; 0 ; 0 ; ; ; 0 ; ; ; 0 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; 0 ; ; ; ; 0 ; ; ; 0 ; ; ; 0 0 ; ; ; ; ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; 0 ; ; ; 0 ; ; 0 0 ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; ; 0 ; 15

16 A1 A2 A3 A4 Appenix C(ont): Orer 44 (Sequenes with zero perioi utoorreltion funtion) Design 22) (13 (13 24) (13 25) (13 26) (13 28) (13 31) (14 15) (14 17) (14 23) (14 24) (14 25) (14 28) ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; 0 ; 0 0 ; 0 ; ; ; ; ; ; ; ; 0 ; 0 ; 0 0 ; ; ; ; ; 0 ; 0 ; ; ; 0 ; ; ; 0 0 ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; 0 ; ; ; ; 0 ; ; 0 ; ; ; ; 0 ; ; 0 ; ; ; ; ; 0 0 ; 0 ; 0 ; ; ; ; ; 0 ; ; 0 ; ; ; 0 ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; 0 ; 0 ; 0 ; ; ; 0 ; ; ; ; 0 ; ; 0 ; 0 ; ; ; ; ; ; ; 0 0 ; ; ; ; ; ; ; ; ; ; ; ; 16

17 A1 A2 A3 A4 Appenix C(ont): Orer 44 (Sequenes with zero perioi utoorreltion funtion) Design 30) (14 (15 22) (15 23) (15 24) (15 27) (15 29) (17 21) (17 22) (17 24) (17 25) (18 19) (18 21) ; ; ; ; ; ; ; ; ; ; ; ; 0 ; 0 0 ; ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; 0 ; 0 ; ; 0 ; ; 0 ; ; 0 0 ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; 0 ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; 0 ; ; 0 ; ; 0 ; ; ; ; 0 0 ; 0 ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 0 0 ; ; ; 0 ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; 0 ; ; 0 ; 0 ; ; ; 17

18 A1 A2 A3 A4 Appenix C(ont): Orer 44 (Sequenes with zero perioi utoorreltion funtion) Design 23) (18 (19 22) (19 23) (20 21) (21 23) ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 0 ; ; ; ; ; ; 0 ; ; ; ; 0 ; ; ; ; ; 0 ; ; ; ; 0 ; ; ; 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 18

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