HADAMARD MATRICES OF ORDER 36 WITH AUTOMORPHISMS OF ORDER 17. are called equivalent if there exist monomial matrices P, Q with PH X Q = H 2

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1 V. D. Tonchev Nagoya Math. J. Vol. 4 (986), 63-4 HADAMARD MATRICES OF ORDER 36 WITH AUTOMORPHISMS OF ORDER VLADIMIR D. TONCHEV Dedicated to Professor Noboru Ito on his 6th birthday. Introduction A Hadamard matrix of order n is an n by n matrix of l's and Γs such that HW = nl. In such a matrix n is necessarily, or a multiple of 4. Two Hadamard matrices H^ and H are called equivalent if there exist monomial matrices P, Q with PH X Q = H. An automorphism of a Hadamard matrix H is an equivalence of the matrix to itself, i.e. a pair (P, Q) of monomial matrices such that PHQ = H. In other words, an automorphism of if is a permutation of its rows followed by multiplication of some rows by, which leads to reordering of its columns and multiplication of some columns by. The set of all automorphisms form a group under composition called the automorphism group (Autίf) of H. For a detailed study of the basic properties and applications of Hadamard matrices see, e.g. [], [7, Chap. 4], [8]. The equivalence classes of Hadamard matrices of orders not exceeding has been determined by Hall [5], [6], More recently, Ito et al. [] completed the classification of the Hadamard matrices of order 4. For the next order 8, it is known that the only primes which can divide the group order are 3, 7, 3 and, and the matrices possessing automorphisms of order 3 or 7 are found [], [8]. A rough but susceptible lower bound for the number of equivalence classes of Hadamard matrices of order 3 can be obtained from a result of Norman [3], stating that there are at least 6689 non-isomorphic Hadamard -(3,5, 7) designs, and hence at least 6689/3 inequivalent Hadamard matrices of order 3. Hadamard matrices of order 36 have been extensively studied [], [], []. It was proved by Ito [9] that if a Hadamard matrix H of order n possesses a "known" doubly-transitive automorphism group then either H Received September 9,

2 64 VLADIMIR D. TONCHEV is of quadratic-residue type, or n = 36. A Hadamard matrix of order 36 with a doubly-transitive group was recently constructed by Ito and Leon []. In this paper we begin a classification of Hadamard matrices of order 36 by means of "local" properties of their automorphism groups, namely, by considering the possible primes dividing the group order. If p is an odd prime dividing the group order of a Hadamard matrix of order n (n ^> 4), then either p divides n or n, or p <L n/ ; moreover, if p does not divide n, then p must be an order of an automorphism of a symmetric -(n, n/, nja ) design []. In particular, the largest prime which can divide the group order of a Hadamard matrix of order 36 is. The Paley matrix [4], [7], which is undoubtedly the first Hadamard matrix of order 36 ever found, admits an automorphism of order. It is perhaps worth noting that none of the 8 Hadamard matrices arising from Steiner triple systems of order 5 [] possesses automorphisms of order. It is our aim in this paper to show that up to equivalence there are precisely Hadamard matrices of order 36 with automorphisms of order. From these matrices, only the Paley matrix has a transitive automorphism group (of order 9584 = 7 3 ), while the remaining matrices all have full automorphism groups of order 68. We use the same method as in []. It is not difficult to see that if a Hadamard matrix H of order 36 admits an automorphism of order, then H is equivalent to a normalized matrix of the form rl l, M : l where M is a (,) incidence matrix of a Hadamard -(35,, 8) design with an automorphism of order. Further, a -(35,, 8) design with an automorphism of order has a block, the derived design with respect to which is a cyclic -(, 8, 7) design. Hence one can obtain the Hadamard -(35,, 8) designs with automorphisms of order as embeddings of cyclic -(, 8, 7) designs. The isomorphism classes of cyclic -(, 8, 7) designs are described in [9], A table with their representatives is given in Section. In Section 3 we study the Hadamard -(35,, 8) and 3-(36,8, 8) designs with automorphisms of order. There are precisely isomorphism classes of -(35,, 8) designs, and classes of 3-(36,8, 8) designs

3 HADAMARD MATRICES 65 with automorphisms of order. In the last Section 4 we show that the Hadamard matrices obtained from the non-isomorphic 3-(3δ, 8, 8) designs are pairwise inequivalent.. Cyclic -(, 8, 7) designs Let X = GF() = {,,,, 6} be the point set of a -(, 8, 7) design which is invariant under the cyclic group Z, i.e. under the permutation (,,,, 6). The set of all (I 7 ) 8-element subsets of X is partitioned into 43 orbits under the action of Z lί. We checked by computer that exactly 6 pairs of these orbits form a -(, 8, 7) design. The images of the blocks of a cyclic -design with point set GF(p) (p a prime) under an affine transformation of GF(p) form again a cyclic design isomorphic to the initial design, and by a theorem of Bays-Lambossy (cf. [3, p. 5]) the converse is also true: two cyclic designs on a prime number of points are isomorphic if and only if they are affine equivalent. The set of the 6 cyclic -(, 8, 7) designs is thus divided into isomorphism classes: classes consisting of 6 designs each, and one class of a single design. Representatives for these designs are listed in Table. Table. Cyclic -(,8,7) designs No. Representatives of block orbits Group order Γ (,,3,4,5,7,,4), (,,3,5,8,,3,4) (,,3,4,5,7,,4), (,,3,7,8,,3,6) 3 (,,3,4,5,8,9,3), (,,4,6,9,,,6) 4 (,,3,4,5,8,9,3), (,,4,8,,,4,6) 5 (,,3,4,5,8,,4), (,,3,6,8,9,,5) 6 (,,3,4,5,8,,4), (,,3,6,,,3,5) 7 (,,3,4,6,7,,3), (,,3,5,6,9,,4) 8 (,,3,4,6,7,,4), (,,4,5,8,9,,3) 9 (,,3,4,6,9,,5), (,,3,5,6,,,6) (,,3,4,6,9,,5), (,,3,5,9,,5,6) (,,3,5,,3,4,5), (,,4,5,7,,,6) 4. The full automorphism groups of these designs can be found with the help of Sims' table of primitive permutation groups of degree [lδ], or by computer using an algorithm of Gibbons [4]. The design has the affine group of GF(Π) as a full automorphism group, while the groups of the remaining designs are all of order. For more details see [9].

4 66 VLADIMIR D. TONCHEV 3. Hadamard -(35,,8) and 3-(36,8,8) designs with automorphisms of order Let D be a symmetric -(35,, 8) design with point set {,,, 35}, and an automorphism β of order. We can assume that /3 = (,,, ) (8) (9,,, 35), and the blocks are labeled in such a way that the block fixed by β is the last one and consists of the points,,,, and the fixed point 8 occurs in the first blocks. Then D has an incidence matrix of the form M N ()... o P Q : όj where M, M, P, Q are circulant matrices, () (M,N) is an incidence matrix of a cyclic -(, 8, 7) design, and... (3) P Q I is an incidence matrix of a -(8, 9,8) design with an automorphism of order. Similarly, (4) ri l o oi is an incidence matrix of a -(8, 9, 8) design with an automorphism of order, and ( 5 ) (AT, P ι ) is an incidence matrix of a cyclic -(, 8, 7) design. Having a quadruple of designs with incidence matrices ()-(5), we can obtain a symmetric design (if one exist) by fixing M, N, P and permuting the rows of Q cyclically while the matrix () produces a design. Let us note that the -(8, 9,8) design with an automorphism of order are easily derived from the cyclic -(, 8, 7) designs: if () is an incidence matrix of a cyclic -(, 8, 7) design, then

5 HADAMARD MATRICES 67 ro... o... l i r i. -. l o... o i [j-m N J> [ M J-N\> where J is the all-one matrix, are both incidence matrices of -(8, 9, 8) designs with an automorphism of order. Conversely, if (3) is an incidence matrix of a -(8, 9, 8) design with an automorphism of order, then (J P, Q) is an incidence matrix of a cyclic -(, 8, 7) design. Another more general way to embed the cycilc -(, 8, 7) designs into symmetric -(35,, 8) designs is to use the algorithm from [6], based on the observation that if a part (i.e. several rows) of the incidence matrix of a -(v, k, X) design is given, then any missing row lies in the orthogonal complement of the vector space over GF(p) generated by the given rows, where p is a prime dividing λ. Let us remark that if (3) is an incidence matrix of a -(8, 9, 8) design completing the design () to a symmetric design (), then the complementary design of (3), i.e. the design with incidence matrix ro... o l... li [J-P J-Q\ also completes the design () to a symmetric design. Therefore, if a -(, 8, 7) design admits an embedding it admits at least two (not necessarily nonisomorphic). It turns out that each of the cyclic -(, 8, 7) designs is embeddable in exactly two symmetric designs. Since the design admits an automorphism of order interchanging the two cyclic block orbits, the symmetric designs thus obtained are given in Table, where in all cases a block JB 35 = {,,, } should be added. Table. Hadamard -(35,, 8) designs with automorphisms of Design D A D B D A D B D ZA D$ B Base blocks B = (,,3,4,5,7,,4,8,,,3,4, 8,9,3,34) i8= (,,3,5,8,,3,4,9,5,6,7,9,3,3,33, 34) B ι = (,,3,4,5,7,,4,9,,5,6,7,3,3,33,35) J5 8 = (,,3,5, 8,,3,4,8,,,,3,4,8,3,35) B x = (,,3,4,5,7,,4,8,,3,4,6,9,3,34,35) B 8 = (..,3,7,8,,3,6,9,5,6,7,9,3,3,33,34) B = (,,3,4,5,7,,4,9,,,5,7,8,3, 3,33) B 8 = (,,3,7,8,,3,6,8,,,,3,4,8,3, 35) B ι = (,,3,4,5,8,9,3,8,9,,3,4,6,3,3,33) B w = (..,4,6,9,,,6,9,5,6,7,8,3,3,3,35) B x = (,,3,4,5,8,9,3,,,5,7,8,9,3,34, 35) #8= (,,4,6,9,,,6,8,,,,3,4,9,33,34)

6 68 VLADIMIR D. TONCHEV Table. (Continued) Design Base blocks D 4A B = (,,3,4,5,8,9,3,8,,3,4,6,8,3,3,34) #iβ= (,,4,8,,,4,6,9,5,6,7,8,3,3,3,35) D, B #i = (,,3,4,5,8,9,3,9,,,5,7,9,3,33,35) #iβ= (,,4,8,,,4,6,8,,,,3,4,9,33,34) D' θa B Ύ = (,,3,4,5,8,,4,8,9,,3,4,7,3,33,34) #iβ= (,,3,6,8,9,,5,9,5,6,7,9,3,3,33,35) D bb B = (,,3,4,5,8,,4,,,5,6,8,9,3,3,35) i8= (,,3,6,8,9,,5,8,,,,3,4,8,3,34) D 6A # x = (,,3,4,5,8,,4,8,9,,3,4,7,9,3,3) #8= (,,3,6,,,3,5,9,5,6,7,9,3,3,33,35) D QB #I =(,,3,4,5,8,,4,,,5,6,8,3,33,34,35) J?i8= (,,3,6,,,3,5,8,,,,3,4,8,3,34) D 7A Bι = (,,3,4,6,7,,3,8,9,,,3,7,9,3,35) #iβ= (,,3,5,6,9,,4,9,4,5,6,7,9,3,3,33) D 7B #i = (,,3,4,6,7,,3,,4,5,6,8,3,3,33,34) i8= (,,3,5,6,9,,4,8,,,,3,8,3,34,35) D SΛ B x = (,,3,4,6,7,,4,8,,,3,6,7,9,3,35) #8= (,,4,5,8,9,,3,,,,3,6,3,3,33,35) D BB #i =(,,3,4,6,7,,4,9,,4,5,8,3,3,33,34) #8= (,,4,5,8,9,,3,8,9,4,5,7,8,9,3,34) D 9A B λ =(,,3,4,6,9,,5,8,,3,8,9,3,3,33,35) #8= (,,3,5,6,,,6,,,,4,6,7,9,34,35) D 9B #!= (,,3,4,6,9,,5,9,,,4,5,6,7,3,34) #8= (,,3,5,6,,,6,8,9,3,5,8,3,3,3,33) D A #i = (,,3,4,6,9,,5,8,9,4,5,9,3,3,33,35) #8= (,,3,5,9,,5,6,,,,4,6,7,9,34,35) DIOB # =(,,3,4,6,9,,5,,,,3,6,7,8,3,34) #8=(,,3,5,9,,5,6,8,9,3,5,8,3,3,3,33) Dπi #i =(,,3,5,,3,4,5,8,9,,,4,8,9,33,35) #8=(,,4,5,7,,,6,9,3,4,8,3,3,3,33,34) In order to establish the nonisomorphism of the designs from Table, we count the number m* of pairs of points occuring together with a given point in precisely ί blocks ( <: i ^ 8). Of course, it is sufficient to do this only for a triple of points belonging to different β-orbits. The results are given in Table 3. Table 3. Design (m, m lt, m 6 ) o? joints D u a l desi^n D Λ

7 HADAMARD MATRICES 69 Table 3. (Continued) Design (m,m l9, m 6 ) points Dΐlal desi^n D B D 3A D B D ZA D 3B D ia D ib D 5A D 5B D 6A D 6B D 7A D 7B D 8A D 8B

8 VLADIMIR D. TONCHEV Table 3. (Continued) Design D Q D A DU (m, wii,,w *β) Number of points 34 Dual design It follows from Table 3 that the designs from Table are all pairwise non-isomorphic with possible exceptions for the pairs (D A, D A ) and (D B9 D B ). However, D ίa and D A (as well as D ίb and D B ) can not be isomorphic since an isomorphism must map the fixed block 35 of D Λ onto the block 35 of D A, and consequently, the cyclic -(, 8, 7) design to the design, which is impossible. The characteristics (m Q,, m 6 ) of the dual designs (i.e. those having as incidence matrix the transpose of the matrix of the initial design) show that D na is self-dual, and the pairs (D ZA, D, A \ (D SB, D ib ), (D 5A, D βλ ), (D 55, D 6B ), (D ΊA, D ΊB ), (D BA, D 8B ), (D 9A, D ίa ), (D gb9 D WB ) consist of designs which are dual to each other. A comparison of the derived cyclic -(, 8, 7) designs in the duals of D ίa, D A, D B, D B shows that (D ίa, D A ) and (D B, D B ) are dual pairs. The data from Table 3 shows also that all designs but D na have full automorphism groups of order. Since each automorphism of D UA must fix the block 35, the automorphism group of D na must be isomorphic to a subgroup of the group of the cyclic -(, 8, 7) design. In fact, the full group of D UA is of order 8-, and is generated by β and the permutation c = ()(, 3, 5, 9,, 6,4, )(4, 7, 3, 8, 5,, 6, ) (8)(9, 5,, 7, 4, 35, 3, 33)(, 9, 8, 6,, 3, 3, 34)(3). Every Hadamard -(4t + 3, t +, t) design is extendable in exactly

9 HADAMARD MATRICES one (up to isomorphism) way to a Hadamard 3-(4t + 4, t +, t) design by enlarging all blocks with a new point, and adding At + 3 new blocks being the complements of the old blocks. Consequently, two 3-(4 + 4, t +, ί) designs are isomorphic iff they possess a pair of isomorphic derived -(4t + 3, t +, t) designs. Moreover, the stabilizer of a point in the automorphism group of a 3-(4t + 4, t +, ί) design ίj coincides with the automorphism group of the derived -(4Z + 3, t +, ί) design with respect to this point, and two points of E are in the same orbit iff the derived designs with respect to these points are isomorphic. If E is a 3-(3β, 8, 8) design being an extension of a -(35,, 8) design with an automorphism β of order, then β acts on E by fixing the new point. Thus at least two of the derived designs of E have automorphisms of order. It is readily seen that a pair of designs (D ia, D ιb ) is extended to isomorphic 3-(36,8, 8) designs, so we have at most non-isomorphic 3-(36,8, 8) designs with automorphisms of order. We denote the extension of D ίa by E t. The characteristics (τz, n u, n Ί ) for the extended designs, where n t is the number of triples of points occuring together with a given point in exactly i blocks, are listed in Table 4, and they show that the 3-designs are non-isomorphic. Table 4. Design (wo, -,nγ) E λ E E E E Number of points

10 VLADIMIR D. TONCHEV Table 4. (Continued) Design #6 E Ί E 8 E 9 -EΊo E n (n Of -,n 7 ]) Number of points 34 Since two points lying in the same orbit under the automorphism group must have identical characteristics, it follows from Table 4 and the preceding comments that the only designs which might have automorphism groups larger than Z ί7 are E 7 and E n. But the derived designs of E 7 with respect to the points having identical characteristics (,, 374, 44, 39, 74,34,) are isomorphic to D 7A and D ΊB respectively, which are nonisomorphic. Hence the full group of E Ί is of order. In the case of E n the derived designs with respect to points 8 and 36 (with characteristics (,, 544,, )) are isomorphic, whence \AutE n \ = AutD nλ \ = The Hadamard matrices Let M be a (,) incidence matrix of a -(35,, 8) design with an automorphism β of order. Then bordering M with a column and row of Γs one obtains a Hadamard matrix on which β acts by fixing the allone row and column. Two Hadamard matrices obtained from symmetric designs, which are extendable to isomorphic 3-designs are equivalent. More precisely, given a Hadamard matrix H (/ι i; ) of order n At + 4 and given k ( k n\ we obtain a 3-(4* + 4, t +, t) design E k = E\H)

11 HADAMARD MATRICES 3 with point set P = {,,, n} and block set {B l9, B k _ u B k+u, B n, B u ", B k _ u B k+u, B n ) where B, = {i: h tj = h ίk ] and B 3 = P - B,. The designs E k (H) and E m (H) are isomorphic iff columns and m oΐ H lie in the same orbit of Aut if more generally, E k (H^) and E k (H ) are isomorphic iff i?ί and iϋγ are equivalent under a signed permutation mapping the column k of H x to column m of H. Moreover, the automorphism group of E k (H) acting on points is permutation isomorphic to the stabilizer in Aut H of column k, acting on signed rows. The automorphism group of any Hadamard matrix contains a subgroup of order generated by ( I, I) which fixes all rows and columns. Hence the order of the automorphism group is Aut if = \AutE k (H)\l k, where l k is the length of the orbit of column k under Aut if []. The Hadamard matrices obtained from the eleven -(35,, 8) designs DiA, * i DUA can be distinguished by the characteristics (n, -, n 7 ) of the related 3-designs E k. Evidently in such a matrix the 3-designs obtained with respect to the columns fixed by β are isomorphic. The matrix corresponding to D A, which is easily seen to be equivalent to the Paley matrix, has the property that its 3-designs E k ( <; k < 36) all have identical characteristic sets, and further analysis shows that all these 3-designs are isomorphic to E n. Hence the order of the group of the Paley matrix is 36-7 = In the remaining matrices, the columns are divided into three orbits; two of length, and one of length. Therefore, all these matrices have groups of order -- = 68. REFERENCES [ ] Th. Beth, D. Jungnickel, H. Lenz, Design theory, Bibliographisches Institut Mannheim/Wien/Zurich, 985. [ ] F. C. Bussemaker and J. J. Seidel, Symmetric Hadamard matrices of order 36, Ann. N.Y. Acad. Sci., 5 (97), [ 3 ] M. J. Colbourn and R. A. Mathon, On cyclic Steiner -designs, Ann. Discrete Math., 7 (98), [ 4 ] P. B. Gibbons, Computing techniques for the construction and analysis of block designs, Technical Report, No. 9, Univ. of Toronto, 976. [ 5 ] M. Hall, Jr., Hadamard matrices of order 6, Jet Propulation Laboratory Research Summary No. 36-, (96), -6. [ Q ] f Hadamard matrices of order, Jet Propulation Technical Report No. 3-76, 965. [ 7], Combinatorial Theory, Gion (Blaisdell), Boston, 967. [ 8 ] A. Hedayat and W. D. Wallis, Hadamard matrices and their applications, Ann. Statist., 6 (978),

12 4 VLADIMIR D. TONCHEV [ 9 ] N. Ito, Hadamard matrices with "doubly transitive" automorphism groups, Arch. Math., 35 (98), -. [] and J. S. Leon, An Hadamard matrix of order 36, J. Combin. Theory, A 34 (983), [], J. S. Leon and J. Q. Longyear, Classification of 3-(4,, 5) designs and 4- dimensional Hadamard matrices, J. Combin. Theory, A 3 (98), [] W. M. Kantor, Automorphism groups of Hadamard matrices, J. Combin. Theory, 6 (969), [3] C. W. Norman, Non-isomorphic Hadamard designs, J. Combin. Theory, A (976), [4] R. E. A. C. Paley, On orthogonal matrices, J. Math. Phys. MIT, (933), 3-3. [5] C. C. Sims, Computational methods in the study of permutation groups, in "Computational Problems in Abstract Algebra" (J. Leech, Ed.), pp , Pergamon, Elmsford, N.Y., 97. [6] V. D. Tonchev, Quasi-residual designs, codes and graphs, Colloq. Math. Soc. Janos Bolyai, 37 (98), [], Hadamard matrices of order 8 with automorphisms of order 3, J. Combin. Theory, A 35 (983), [8], Hadamard matrices of order 8 with automorphisms of order 7, J. Combin. Theory, Ser. A, 4 (985), 6-8. [9], R. V. Raev, Cyclic -(, 8, 7) designs and related doubly-even codes, Compt. rend. Acad. bulg. Sci., 35 (98), [] W. D. Wallis, Hadamard equivalence, Congressus Numerantium, 8 (98), 5-5. Institute of Mathematics Sofia 9, P.O. Box 373 Bulgaria