Biometrics Unit, 337 Warren Hall Cornell University, Ithaca, NY and. B. L. Raktoe

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1 NONISCMORPHIC CCMPLETE SETS OF ORTHOGONAL F-SQ.UARES, HADAMARD MATRICES, AND DECCMPOSITIONS OF A 2 4 DESIGN S. J. Schwager and w. T. Federer Biometrics Unit, 337 Warren Ha Corne University, Ithaca, NY and B. L. Raktoe Department of Economics and Statistics Nationa University of Singapore Kent Ridge Singapore, 0511, Maay BU-807-M February, 1983 Key Words and Phrases: F-square design; Hadamard product; factoria design; atin square

2 NONISCMORPIITC CCMPLETE SETS OF ORTHOGONAL F-SQUARES, HADAMARD MATRICES, AND DEC<4POSITIONS OF A zi DESIGN S. J. Schwager and w. T. Federer B. L. Rak.toe Biometrics Unit 337 Warren Ha Corne University Ithaca, NY Dept. of' Econ. and Statist. Nationa University of' Singapore Kent Ridge Singapore, 0511, Maay ABSTRACT The f'ive nonisomorphic Hadamard matrices of' order 16 given by Ha (1961) correspond to f'ive distinct decompositions of' a 24 design into singe-degree-of'-f'reedom orthogona contrasts. The anaysis of' variance tabes associated with these decompositions are compared, with emphasis on the three cases that have a 4 x 4 row and coumn structure. The three corresponding casses of' Hadamard matrices generate nonisomorphic compete sets of' nine orthogona F(4;2,2)-squares, one of' which shows a previousy unreported pattern; the remaining two Hadamard casses do not produce compete sets of' F-squares. 1. INTRQWCTION A square matrix H of' order n is a Hadamard matrix if' each of' its entries is + or -1 and each pair of' its rows is orthogona, that is, HH' = ni For a genera treatment of' Hadamard matrices,

3 see Hedayat and Wais (1978). A Hadamard matrix is normaized if its first row and coumn consist entirey of +'s. Two Hadamard matrices are isanorpbic (or equivaent) if one can be obtained fran the other by a sequence of row and coumn permutations and row and coumn canpementations. Ha (1961) derived five nonisomorphic Hadamard matrices of order 16 and showed that any Hadamard matrix of order 16 must be isomorphic to one of these five. The set of a Hadamard matrices of order 16 therefore consists of five equivaence casses. The representatives H to H 5 of these five casses given by Ha are reproduced in Tabe I. The symbos + and - respectivey, as they wi throughout this paper. represent the numbers + and -, Thus, for exampe, the compement of a row of a Hadamard matrix is obtained by muti pying each eement of the row by -. The rest of this paper is organized as foows: Section concudes with a brief discussion of Hadamard products, incuding two emmas concerning rows of +'s and - 's, and F-squares. There is a natura one-to-one correspondence reating the equivaence casses of Hadamard matrices of order 16 and the decanpositions of a ~ design into singe-degree-of-freedom orthogona contrasts. This correspondence is presented in Section 2. Tb.e three decompositions associated with ANOVA (anaysis of variance) tabes having a usefu row and coumn structure to be defined in Section 2 are treated in detai. In Section 3, it is shown that the three corresponding Hadamard casses produce nonisomorphic compete sets of nine orthogona F(4;2,2)-squares, one of which is not isomorphic to any previousy produced set. Section 4 estabishes that the remaining two Hadamard casses do not produce compete sets of F-squares. Section 5 contains concuding remarks. The Hadamard product, eement-by-eement mutipication of equay-dimensioned vectors or matrices, wi be denoted by *. For a genera discussion of this operation, see Seare (1982, Sec. 2.8). The identity eement for Hadamard mutipication of rows is the row ~ = ( ) A set of row vectors is cosed under *

4 TABLE I Nonisomorphic Hadamard Matrices o:f Order 16 Given bz Ha a. ~ b. ~ c. H d. H4 e. H5 if the Hadamard product o:f any two vectors in the set is aso contained in the set. A cosed tripe is a set S o:f three nonidentity rows such that S U (~} is cosed. Cosed tripes are strongy reated to the tripes defined :from bock design considerations and tabed :for 1_ to :H 5 by Ha ( 1961). If he had used row numbers

5 instead of coumn numbers to represent the bock design objects, his tripes woud be identica to the cosed tripes treated here. The foowing two emmas appy to rows of any ength. They wi be usef'u in Sections 2 and 4. The first simpifies the process of finding cosed tripes of rows. The second indicates that there is a specia reationship between the row e and cosed tripes. Lemma.. The set [~'~2,~ 3 } of three distinct rows whose eements are +'s and-'s is a cosed tripe iff (if and ony if) ~ *~2= ~3. Proof. The Hadamard product of any pair of these rows equas the remaining row, for instance, ~ * ~ 3 = ~ * ( ~ * ~2 ) = ~ * ~2 = ~2 Q,ED Lemma.2. If three orthogona rows whose eements are +'s and - 's formacosed tripe (~'~2,~ 3 }, then each of these rows is orthogona to ~' -1 s. Proof. that is, each row consists of equa numbers of +'s and Let n denote the row ength. Let #( + + +) denote the number of row positions where rows ~' ~2, and ~ 3 contain +, +, and +, respectivey; et 41=( + + -) denote the number of positions where ~'!" 2,! 3 contain +, +, -; and so on. By the ast emma,! * ~ 2 = ~ 3, so #(++-) = 41=(+-+) = #(-++) = 41=(---) = 0 ' 4f( +++) + #( --+) + #( -+-) + 41=( +--) = n. The orthogonaity of : to ~2, : to :3' and ~ 2 to : 3 yieds 41=( +++) + #(--+) - 41=(-+-) - 41=(+--) = 0 'If(+++) - 41=(--+) + #(-+-) - 41=(+--) = 0 41=(+++) - 41=(--+) - #(-+-) + 41=(+--) = 0 ' ' Soving the ast four equations gives #(+++) = 41=(--+) = 41=( -+-) = #( +--) = in. The number of +' s in ~ equas 41=( +++) + =If(+--) = jn, and ~2 and ~ 3 are treated simiary. QED

6 An F(2X.; A., }.)-square with two treatments ~ symbos or simpy an F-square is a 2A. x 2}.. matrix in which every row and coumn ccntains each of the two symbos exacty A. times o Two of these F squares are orthogona if each possibe ordered pair of symbos appears together }..2 times when one of the F-squares is superimposed on the other. A set of s F-squares is mutuay orthogona if every pair of these s F-squares is orthogona. Such a set is denoted by OF(2}..; A., X.; s), and is compete when s = (2>..-1) 2 These definitions suffice for the presentation in this paper. More genera definitions and discussion of the practica appication of F-square designs are found in Hedayat and Seiden (1970) and Hedayat, Raghavarao, and Seiden (1975). 2. HADAMARD CLASSES AND 1-DoFo ANOVA TABLES The standard anaysis of a 2 4 design invoves a grand mean M, main effects A,B,C,D, two-factor interactions AB,AC,AD,BC,BD,CD, three-factor interactions ABC,ABD,ACD,BCD, and four-factor interaction ABCDo These 16 terms for a coumn vector ~ = [2M,A,B,AB,C,AC,BC,ABC,D,AD,BD,ABD,CD,ACD,BCD,ABCD]', where the first entry of ~ is twice the grand mean M o A other entries wi be referred to simpy as effects. Using standard notation, define the vector of the 16 treatment combination means or ce means v - [(abed), (abc), (abd), (ab), (acd), (ac), (ad), (a), (bed), (be), (bd), (b), (cd), (c), (d), (1)]', where (abed) denotes the mean of a observations receiving the treatment combination abed, and so on. Then ~ 1 can be obtained from! by 1 ~1 = B II:t_Y ' where the Hadamard matrix II:t_ appears in Tabe Iao The orthogonaity of H1 estabishes easiy the we-known fact that the ast 15

7 eements of ~ form a set of mutuay orthogona singe-degree-offreedom ccntrasts. These contrasts partition the degrees of freedan in a ~ design. 'Ihe 16 rows of If:J_ form a cosed set. Tabe IIa gives the number of the row obtained as the Hadamard product of each pair of nonidentity rows of If:J_, for exampe, row 2 *row 3 =row 4 Tabe IIIa ists the 35 cosed tripes, which are easiy checked with the hep of Tabe IIa and Lemma.; five disjoint cosed tripes are starred. (Other choices of the five are possibe, giving different but isomorphic resuts. The seection made here wi faciitate the comparison of ANOVA tabes reated to IS_, f2, and ~ ) Repace the row numbers in these tripes by the corresponding entries of~: and so on. for exampe, 2,9,0 by A,D,AD; 3,5,7 by B,C,BC; The resut is five sets of three effects. Each set has the important property: members is the set's remaining member 1 mutipication This division of the 15 effects of ~ members is the set's remaining member. (P) the interaction of any two of its This division of the 15 nonmean effects of ~ i.e., it is cosed under into these five into these five sets of three effects produces the.anova tabe shown in Tabe IVa. It is reated to the Factoria Compete Confounding Construction of Federer et a. (1969), which has been used to construct OL(4, 3), the compete set of three orthogona atin squares of order 4. Take a 2" factoria experiment with factors a, b, c, and d and write the treatment canbinations in a 4 X 4 square whose rows are confounded with main effects A, D, and their interaction AD and whose coumns are confounded with main effects B, C, and their interaction BC (This choice of effects is again for convenience, agreeing with two of the starred tripes.) The resut, with 0 and representing the two eves of each factor, is

8 TABLE II Hadamard Products of Nonidenti ty Rows of IS_ to H a. H b.~

9 TABLE II (cant.) c. H d. H

10 TABLE II (cont.) O O e. H O O TABLE III Cosed Tripes of Rows of ~ to H 5 2,3,4 3,14,16 6,10,13 2,3,4 3,14,16 2,3,4 2,3,4 2,3,4 2,5,6 4,5,8 6,,6 2,5,6 4,5,8 2,5,6 2,5,6 2,5,6 2,7,8 4,6,7 *6,2,5 2,7,8 4,6,7 2,7,8 2,7,8 2,7,8 *2,9,10 4,9,2 7,9,15 *2,9,10 4,9,2 2,9,10 3,5,7 2,9,10 2,,2 4,O, 7,10,16 2,,2 4,O, 2,,2 3,6,8 2,,2 2,13,14 ~,3,6 7,,3 2,13,14 4,3,6 2,13,14 4,5,8 2,13,14 2,15,16 4,14,15 7,2,4 2,5,6 4,14,15 2,15,16 4,6,7 2,15,16 *3,5,7 5,9,13 8,9,16 *3,5,7 3,5,7 3,6,8 5,O,4 8,O,5 3,6,8 3,6,8 3,9, 5,,5 8,,4 3,9, 4,5,8 3,O,2 5,2,6 8,2,3 3,O,2 4,6,7 3,13,15 6,9,14 3,13,15 a. 1_ b. ~ c. H 3 d. H4 e. H 5

11 TABLE BD IV ANOVA Tabes Obtained f'rcm H1 to H 3 Source of' Variation Degrees of' Freedcm a. ~ AN OVA Tabe Mean Rows 3 A D AD Coumns 3 B c BC Latin sq_uare treatments 3 AB CD ABCD Latin sq_uare 2 treatments 3 AC ABD BCD Latin square 3 treatments 3 ABC ACD Tota 16 b. ~ ANOVA Tabe Mean Rows 3 A D AD Coumns 3 B c BC Cycic atin square treatments 3 AB G21 G24 AC 1 ABC BD ABD G22 G23 Tota 16

12 TABLE IV (cont.) c. ~ ANOVA Tabe Source of' Variation Degrees of' Freedom Mean Rows 3 A D AD Coumns 3 B c BC AB AC ABC G31 G32 G33 G34 G35 G36 Tota 16 (B)o, (C)o (BC) 0 (B) '(c)o (BC) (B)o' ( c) (BC) 1 (B),(c) (BC) 0 (A) 0, (D)0, (AD)0 [ b c be (A\, (D) 0, (ADh a ab ac abc (A) 0, (D\, (AD) 1 d bd cd bed (A)' (D)' (AD)o ad abd acd abed ] 0 (2.1) The nine interaction ef'f'ects of' ~ that are not confounded with rows and coumns f'orm three more tripes with property (P): AB, CD,ABCD; AC,ABD, BCD; and ABC, BD,ACD. Confounding symbos I to IV with each of' these tripes in turn as rows were confounded with A, D,AD, f'or exampe,

13 I: (AB) 0, (CD) 0, (ABCD) 0 II: (AB) 0, (CD) 11 (ABCD) 1 III: (AB) 1, (CD) 0, (ABCD) 1 IV: (AB) 1, (CD) 1, (ABCD) 0 ' and inserting these symbos into the appropriate ces of (2.1) produces three orthogona atin squares. The five sets of three effects in Tabe IVa thus can be associated with rows, coumns, and three orthogona atin squares. A 16-degree-of-freedom ANOVA tabe has ~and coumn structure if it contains a grand mean effect and at east two disjoint sets of three has row and coumn structure if its rows incude e and at east two disjoint cosed tripes. This structure wi have an important roe in the treatment of compete sets of F-squares in Sections 3 and 4. treatment of compete sets of F-squares in Sections 3 and 4. The Hadamard matrix H transforms! into _the vector ~ 1, whose eements are effects associated with one-degree-of-freedom rows in the standard ANOVA tabe. Any Hadamard matrix isomorphic to H produces the same ANOVA tabe except for permutation of the symbos A, B, C, D The use of Hadamard matrices not isomorphic to H wi now be examined. Using ~ from Tabe I, define the vector ~ = ~2! As with~' the orthogonaity of ~ estabishes that the ast 15 eements of ~ are mutuay orthogona singe-degree-of-freedom contrasts. The first tweve rows of H and of ~ agree, so ~ = ~[2M, A, B,AB, C,AC, BC,ABC, D,AD, BD,ABD, ~1' G22' G23' G24] I ' where matrix agebra shows that ~1 = j( CD +ACD +BCD- ABCD) G22 = j-( CD +ACD- BCD +ABCD) ~3 = j-( CD- ACD +BCD +ABCD). ~4 = j(-cd+acd+bcd+abcd)

14 The rows of ~ do not form a cosed set. Tabe IIb gives the row number of Hadamard products of rows of ~' with 0 indicating that this product is not a row of ~' as is the case for row 5 *row 9. There are 19 cosed tripes, which are isted in Tabe IIIb; they are easiy found by using Lemma 1.1. Three disjoint cosed tripes are starred, and no arger set of disjoint cosed tripes can be found. (As with H' other choices are possibe.) Repacing the row numbers in these tripes by the corresponding entries of ~ produces the three sets of three effects A,D,AD; B,C,BC; and AB, G21, G24. Each set has property (P), and no further sets with this property can be formed from the remaining six effects. This division of the fifteen effects of ~ resuts in the ANOVA tabe shown in Tabe IVb, which has row and coumn structure. An isomorphic version of this tabe, with A, B, c, D permuted, was shovm by Mandei (1975) to correspond to a cycic 4 X 4 atin square design. The three sets satisfying property (P) can be associated with rows, coumns, and a cycic atin square. The same approach can be appied to The first 10 rows of H and ~ agree, so ~3 = ~[2M,A, B,AB, C,AC, BC,ABC, D,AD, G31' G32' ' G36] I ' where matrix agebra gives G31 = i'( BD +ABD + CD - ACD) G32 = i ( BD + ABD - CD + ACD ) G33 = t( CD+ACD+BCD-ABCD) G34 = i( CD+ ACD- BCD+ ABCD) G35 = i( BD -ABD+BCD+ABCD) G36 = i( -BD +ABD +BCD +ABCD) The rows of H 3 do not form a cosed set. Tabe IIc gives the Hadamard products of rows of H 3, with 0 indicating a product that is not a row of H 3 Tabe IIIc ists the 11 cosed tripes, of which no more than two are disjoint. Repacing the row numbers in the

15 starred tripes by the corresponding entries of ~ 3 produces the two sets of' three effects A, D,AD; and B, C, BC Each of' these sets has property (P), and no further sets of' this type can be formed. This division of the f'if'teen ef'f'ects of ~ resuts in the ANOVA tabe shown in Tabe IVc, which has row and coumn structure. The two sets satisfying property (P) can be associated with rows and coumns. Simiar singe-degree-of-freedom ANOVA tabes can be derived f'or the vectors Hadamard products of rows and cosed tripes appear in Tabes IId and IIId for H 4 and in Tabes IIe and IIIe for H 5. In neither case can two disjoint pairs of cosed tripes be found. A singe cosed tripe corresponds to a set of three effects that can be associated with either rows or coumns, but not both. 'ius the AN0VA tabes f'or these cases do not have row and coumn structure. This ack has important impications for the F-square methods treated in the next two sections. These AN0VA tabes are straightforward to derive and wi be omitted here. The discussion of' this section is summarized as foow. The nonisomorphic Hadamard matrices H to H 5 correspond to five decompositions of a 2 4 design into a singe-degree-of-freedom ANOVA tabe. Three of' these are given in Tabe IV. 3. HADAMARD CLASSES GENERATING CCMPLETE SETS OF F-SQUARES Assume that a Hadamard matrix H of order 16 has row and coumn structure. Set aside the row e and two disjoint cosed tripes of' rows, which may be associated with row and coumn structure in a 4 x 4 design. If the eements of each of the remaining nine rows of H are suitaby rearranged and used to fi a 4 X 4 matrix, a compete set of' nine orthogona F(4 ;2, 2)-squares is produced. When row and coumn structure is absent, however, the rows of H do not yied a compete set of' orthogona F-squares. These resuts wi be estabi_@ed and appied in this section and the next.

16 For an effect E in ~' et!'e denote the row of ~ for which E = ~E!. For exampe,!'a' ~B' ~C' and ~D denote rows 2, 3, 5, and 9 of IS_, respectivey;!ab and!'en denote rows 4 and 13; ~BC and ~AD denote rows 7 and O Row of ff:t_ is the identity ~ Define the disjoint cosed tripes R =(!A'!'B'!'AB} and Q = (~C'!'D' ~CD} For any row vector ~ = (~ ~ ~6 ), define the 4 x 4 matrix..1(~) whose (i, j)th entry is x4 (i-)+j: x ~ x3 Xt./(~) - x5 X6 x7 X8 ~ x1o xu x2 ~3 x4 x5 x6 Lemma 3-. Let a Hadamard matrix H of order 16 have ~' R, and Q among its rows. Then for any other row!' of H,..I(!) is an F(4;2,2)-sq_uare. When...1(!') is formed for each of the nine remaining rows of H, a compete set of mutuay orthogona F-squares is obtained Proof. The rows of ~' R, and Q are e ~ ~B ++-t+ ++++!A B = (3.1)!'c ~D !'co For any other row!' of H, et n1,n2,n3,n4 be the number of +'s among its first, second, third, fourth set of four eements. Let ~ be the number of +'s among eements, 5, 9, and 13 of ~; ~ the number of +'s among eements 2,6,10,14; m 3 the number of +'s among eements 3,7,,5; and m 4 the number among eements 4,8,12,16. Orthogonaity of r to the rows of e and R yieds

17 e n + ~ + n3 + n4 =8 - '!'A n + n n n4 = 8 ~B n1 +4- n2 + n n4 = 8 ' r -AB: n n n3 + n4 =8. Thus~ =n 2 =n 3 =n4 =2, so each row of'...p(!') has 2+'sand2-'s. Orthogonaity of!' to ~' ~C'!'D' and!'en gives the same equations with n 1,n 2,n 3,n 4 repaced by ~,~,m 3,m 4, has 2 +'s and2- 's, and...p(:) is an F-square. so each coumn of'...p(!') The F-squares...1(!') formed from any two remaining rows of H are orthogona, since the Hadamard product of' these F-squares contains equa numbers of' +'s and - 's. The set of nine F-squares of' this type is compete, because 9 = (4-1) 2 Q;ED Lemma 3.2. If' a Hadamard matrix H of' order 16 has e and at east two disjoint cosed tripes S and T among its rows, then H is isomorphic to a Hadamard matrix whose first seven rows are ~' Q Aso, H is isomorphic to 11._, ~' Proof'. or H 3 R, and Permute the rows of' H so e is the first row, foowed by the three rows of' s, then the three rows of' T, then the remaining nine rows. Next, permute the coumns to make the second row equa to!'a Foow this by permuting the first eight coumns, and then permuting the second eight coumns, to make the third row equa to :B. By cosure of' the tripe, the fourth row must equa :AB. Ca this matrix H a Let n 1,n 2,n 3,n 4 be the number of' +'s among the first, second, third, and fourth set of four eements of' the fifth row of H It a was shown in the ast emma that n 1 = n 2 = n 3 = n 4 = 2, permutation of' coumns 1 to 4 of H, then coumns 5 aowing the to 8, then 9 a to 12, then 13 to 16, to make the fifth row equa to!'c. Ca this matrix Hb. Let m,... ' ms be the number of +Is in the first,..., eighth pair of' eements in the sixth row of fb. The sixth row is orthogona to ~ enma give and R, so the equations in the proof of the ast

18 Simiary, since the seventh row of fb is the product of ~C sixth row by cosure, ~ ~ = m m4 = m m6 = ~ ms = 2 and the Soving these equations shows that ~ = ~ = = m 8 = 1, so the coumns of ~ can be permuted, a pair at a time, to make the sixth row equa to ~D. equa ~CD by cosure. any row that 'begins with - When this has been done, the seventh row must Ca the resuting matrix He. Compement (if H was normaized, none wi occur) and ca the resuting matrix Hd Both He and Hd are isomorphic to the origina H, and both have ~' rows. R, and Q as their first seven Ha (1961) noted that any Hadamard matrix of order 16 whose first row and coumn consist entirey of +'s is equivaent to one of the five matrices H to H 5 under permutation of rows and coumns. (This property does not hod for higher orders, where compementatioo broadens the equivaence casses.) The number of disjoint cosed tripes is invariant under row and coumn permutations. Therefore Hd' which has at east two disjoint cosed tripes, cannot be equivaent to either H 4 or H 5, each of which does not, so Hd must be equivaent to H' f2 1 or H 3. QED Theorem 3.1. I..et H be any Hadamard matrix of order 16 with row and coumn structure. A compete set of orthogona F-squares can be generated from H by performing a series of coumn permutations on H and then appying J to the nine rows not invoved in estabishing the row and coumn structure of H. Proof. Transform H to H as in the proof of Lemma 3.2, and obtain c the set of F-squares by appying J to each of the ast nine rows of H as in Lemma 3.1. c QED 'Ibis method wi now be used on ~' ~' and H 3 Choose cosed tripes S=t~A'!D'!AD} and T=(~B'~C'~BC}' which occur in~'~' and ~ Transform the rows ~'~A' ED' ~AD' ~B' ~C' ~BC' which are rows 1,2,9,10,3,5,7, into the pattern of (3.1) by permuting the coumns, writing them in the order 1,3,5,7,2,4,6,8,9,11,13,15,10,12,14,16.

19 App:cying J to the remaining rows 4, 6, 8, and to 16 of H]_, ~' or E) gives a compete set of orthogona F-squares. These three sets of F-squares are shown in Tabe V. Definition. Two sets of F-squares are isomorphic if the F-squares of one set can be transformed into the F-squares of the other set by a series of row and coumn permutations on each F-square. The set of F-squares obtained frcm HJ. is isomorphic to a compete set of orthogona F(4;2,2)-squares given by Hedayat, Raghavarao, and Seiden (1975). (A minor error appears in their F-square F 3, whose ast two rows must be interchanged.) The setoff-squares obtained from ~ is isomorphic to a compete set given by Mande (1975), and to another compete set given by Schwager, Federer, and Mande (1982). However, the sets off-squares obtained from H' ~' and ~ are not isomorphic to each other; the ast of these is not isomorphic to any compete set reported previousy Theorem 3.2. The three compete sets of orthogona F-squares in Tabe V are nonisomorphic, that is, no two of these sets are isomorphic. Any compete set is isomorphic to one of these three. Proof. The number of distinct row patterns occurring in an F square is invariant under row and coumn permutations of the F-square. Consequenty, row and coumn permutations on each F square in a set cannot change the number of F-squares in the set that have a specified number of row patterns. Each of the nine F-squares derived from H1 contains exacty two distinct row patterns. Two distinct row patterns occur in the first five F-squares derived from ~' row patterns. whie the ast four of the ~ F-squares have four Two row patterns appear in the first three F-squares derived from E)' whie the ast six of the E) F-squares have four row patterns. The invariance of the number of two-row and four-row F-squares in a set impies that none of the three sets from Tabe V can be transform~d into another of these sets by row and coumn permutations. Lemmas 3.1 and 3.2 prove tbe ast pa.rt. QED

20 TABLE v Compete Sets of' Orthogona F-Squares Obtained f'rom 11. to ~ a. 11_: II II u u u b ~: II II II II II c. H3: Equa signs indicate identica F-squares 4. HADAMARD CLASSES THAT FAIL TO PRODUCE F-SQUARES '!be existence of' three nonisomorphic canpete sets of' orthogona F-squares raises the question of' whether H4 and H 5 ead to f'urther sets of' this type. Because H 4 and H 5 ack the row and coumn structure that payed a key roe in the construction of' F squares f'or 11_, H2' and H 3, the answer is negative. This is a consequence of' the f'oowing two theorems. Theorem 4.1. If' a Hadamard matrix H of' order 16 contains the row e and nine rows that become orthogona F-squares when the operator J is appied, then the remaining six rows of' H are, up to row compementation, the rows in the set R U Q.. Proof'. Let : 1,., : 6 denote the remaining rows of' H They are orthogona to each other, to ~' and to a nine of' the rows giving F-squares under J. The six rows in R U Q. = [:A':B' ::AB'!'c' ::D' :"en} aso have these properties. Thus, viewed as vectors in 16-dimensiona

21 space, R U Q and!' 1 to!' 6 span the same subspace, which is the orthogona. compement o:f the span of ~ and the nine F-square rows. The row!' 1 can be written as a inear combination o:f the basis vectors in RU Q, (4.1) It wi now be shown that one o:f the coefficients k. must be ±1 and. the rest must be 0; in other words,!' 1 is either a row in R U Q or the compement of such a row. Consider the vector equation (4.1) eement by eement, using the numerica :forms o:f the basis vectors in R U Q given in ( 3.1) The row!' consists of equa numbers o:f +'s and-'s, that is, eight +'s and eight -'s, so transposing (4.1) gives equation (4.2): ± ± r' - ± [r' r' r' r' r' = -A -B -AB -C -D!'cnJ~ =. ~' ± where ~= [~ ~ ~]'. Let u denote a 6 x1 coumn vector of +1's and -'s; there are '::!3 z possibe choices o:f u. Define w= ' Any soution k of (4.2) must satisfy Zk=U :for scme - ~' as these are rows 1 to 5 and 9 of (4.2). The matrix W is the inverse o:f Z, so -1 k=z U=WU To sove (4.2), substitute every possibe vector~ into ~=Wt_:. For each ~' insert the resuting ~ into the right-hand expression in (4.2). If evauating this matrix product gives a vector ccnsisting o:f equa numbers of +'sand-'s, then~ is a soution o:f (4.2)

22 Direct cacuation (by computer) shows that the ony choices of ~ eading to soutions are the six coumns of Z and their compements. The corresponding vaues of k are thus the six rows of a 6 X 6 identity matrix and the negatives of these rows. This demonstrates that a singe k. is ±1 and the rest are 0. J. Consequenty, ~ must be a row in R U Q or the compement of' such a row. The same reasoning appies to ~ 2,, ~ 6 Because ~ to ~ 6 span the same subspace as R U Q, each row in R U Q must appear, possiby af'ter compementation, exacty once in ( : 1,, ~6 }. QED For a Hadamard matrix of' order 16 containing nine rows that become orthogona F-squares under J, it may not be true that the remaining seven rows are ~ and RU Q, up to row compementations. That is, the assumption that e is a row of' H is necessary in Theorem 4.1. This can be seen by repacing the rows of Q and ~ in any Hadamard matrix whose rows incude ~' R, and Q, for exampe ff:i_, with the rows It is an exercise to verif'y that these rows are orthogona and that they constitute an aternate basis f'or the span of e and Q in 16-space, making them orthogona to the other 12 rows of the matrix. The matrix is thus a Hadamard matrix. By Lemma 3.1, nine of the other 12 rows become orthogona F-squares under s.. This demonstrates the necessity of ~ 's presence in H Theorem 4.2. Let L 1, L 2, and L 3 be disjoint sets of' one, six, and nine row numbers, respectivey, from the coection [1, 2,..., 16} For a normaized Hadamard matrix H of' order 16, any two of' the foowing conditions impy that the remaining one hods:

23 (i) the row ~ of H is ~; (ii) the six rows ~ of H constitute two disjoint cosed tripes; and (iii) the nine rows L 3 of H can be transformed by coumn permutations of H into rows that become orthogona F-squares under Proof.,./. For convenience in visuaizing the structure of H, the partition of [ 1, 2,, 16} may be thought of as L 1 = [ 1}, L2 =(2, 3,..., 7}, and L 3 =(8, 9,, 16}. If ( i) and ( ii) hod, then H has row and coumn structure, and Theorem 3.1 shows that the rows L 3 of H generate a compete set of orthogona F-squares, proving (iii). Assume (i) and (iii). Perform the coumn permutations on H that transform the rows L 3 into rows giving orthogona F-squares under,.i. Theorem 4.1 and the fact that H is normaized impy that the rows ~ of the resuting matrix are the two disjoint cosed tripes R and Q. Cosed tripes are invariant under row and coumn permutations, so the rows ~ of H must aso have been cosed tripes, which proves (ii). Finay, assume (ii) and (iii). The nine rows L 3 of H become F-squares when coumns are permuted and,.i is appied, so each of these rows has eight +'sand eight -'s. Viewed as vectors in 16- space, these rows are a orthogona to ~. By Lemma 1.2, the six rows ~ of H are aso orthogona to e Therefore, the row ~ of H must be e QED If the assumption that H is normaized in Theorem 4.2 is removed, the resut is a itte more cumbersome to state but essentiay unchanged. Coroary 4.1. Let L 1, L2, and L 3 be as in Theorem 4.2. For any Hadamard matrix H of order 16, any two of the foowing impy the third:

24 (i') the row~ of His~ or-~; (ii') the six rows~ of Hare disjoint tripes [~1,~ 2,~ 3 } and ( ~4' ~5' ~6} with ~ * ~2 = ±!'3 and ~4 * ~5 = ~6; and ( iii ) the nine rows L 3 of H can be transformed by coumn permutations of H into rows that become orthogona F-squares under '*' Proof. Assume (i') and (ii'). Repace -~by~ if~ is -~, ~ 3 by -~ 3 if ~ * ~2 = -~ 3, and ~ 6 by -~6 if ~4 * ~ 5 = -~6. Lemma 1.1 demonstrates row and coumn structure of the resuting matrix, whose rows L 3 generate orthogona F-squares by Theorem 3.1, showing (iii). The other two parts are simiar variations of the proof of Theorem 4.2. QED Theorems 4.1 and 4.2 and Coroary 4.1 show the cose connection between row and coumn structure of an order 16 Hadamard matrix H and the existence of a compete set of orthogona F-squares based on the rows of the matrix. If H contains e and nine rows that give orthogona F-squares when J is appied after sui tabe coumn permutations, then H has row and coumn structure, possiby after compementation of one or two rows. Conversey, if H has row and coumn structure, then the nine rows uninvoved in estabishing this structure are transformed into orthogona F-squares by coumn permutations and J. It is an immediate coroary of Theorem 4.2 that H4 and H 5 do not give compete sets of orthogona F-squares when the procedure of Section 3 is appied. This is because such a compete set woud impy the presence of row and coumn structure, which both H4 and H 5 ack

25 5. CONCLUDING REMARKS F-squares Every F-square in Tabe V is isomorphic to one of' the two F - [~~]' F2 = [~~:] Each F-square containing exacty two distinct row patterns can be transformed to F 1, and each F-square with f'our distinct row patterns to F 2, by a series of' row and coumn permutatiqns. The division of' a compete set of' orthogona F-squares into isomorphism c~sses is an important method f'or determining whether severa such sets have basic structura dif'f'erences. For exampe, Tabe Va contains 9/0 F-squares isomorphic to F 1 /F 2, Tabe Vb contains 5/4, and Tabe Vc contains 3/6. The resuts of' Sections 3 and 4 demonstrate that no other pattern can exist f'or any OF(4;2,2;9) set The generaization of' these resuts to F-squares and Hadamard matrices of' higher order is hampered by severa dif':f"icuties. One is the size of' the matrices invoved; Hadamard matrices of' order 64 and sets of' 49 orthogona F(8;4,4)-squares are invoved in the next arger probem, f'oowed by Hadamard matrices of' order 256 and sets of' 225 orthogona F(6;8,8)-squares. Another is that the number of' equivaence casses of' Hadamard matrices of' order 2m increases with m, and is not known even f'or sma vaues of' m. For references, see Hedayat and Wais (1978, p. 1188). BIBLIOGRAPHY Federer, w. T., Hedayat, A., Parker, E. T., Raktoe, B. L., Seiden, E., and Turyn, R. J, (1969). Some techniques f'or constructing mutuay orthogona atin squares. ARO-D Report 70-2, PP Ha, M., Jr. (1961). Hadamard matrices of' order 16. Jet Propusion Lab. Research Summary 36-10, ~ J.P.L., Pasadena, Caifornia

26 Hedayat, A., Raghavarao, D., and Seiden, E. (1975). Further contributions to the theory of F-squares design. Ann. Statist. 3, Hedayat, A. and Seiden, E. (1970). F-square and orthogona F squares design: A generaization of Latin square and orthogona Latin squares design. Ann. Math. Statist. 41, o44. Hedayat, A. and Wais, w. D. (1978). Hadamard matrices and their appications. Ann. statist. 6, Mandei, J. P. (1975). Compete sets of orthogona F-squares. Uhpubished Master's thesis, Biometrics Unit, Corne University. Seare, s. R. (1982). Matrix Agebra Usefu for Statistics. Wiey, New York. Schwager, s. J., Federer, W. T., and Mandei, J. P. (1982). Embedding cycic atin squares of order 2n in a compete set of orthogona F-squares. No. BU-788-M, Biometrics Unit, Corne University, Ithaca, New York