Maximal determinants and saturated D-optimal designs of orders 19 and 37

Maximal Determinan ts and Saturated D-optimal Designs of Orders 19 and 37 Ric hard P . Bren t Mathematical Sciences Institute, Australian National Univ ersit y , Can b erra, A CT 0200, Aus tralia maxdet@rpbr ent.com William Orric k Departmen t of Mathematics , Indiana Univ ersit y , Blo omington, IN 47405, USA worrick@ind iana.edu Judy-anne Osb o rn Univ ersit y of New castle, Callaghan, NSW 2308 , Australia Judy-anne.O sborn@anu.edu.au P aul Zimmermann INRIA Nancy – Gra nd Est, Villers-l ` es-Nancy , F rance Paul.Zimmer mann@inria.fr Octob er 1, 2018 Abstract A saturated D-optimal desig n is a { +1 , − 1 } square matrix of given order with maximal determinant . W e sea r c h for saturated D-optimal designs of orders 19 and 37, and find that known matrice s due to Smith, Cohn, Orrick and Solomon are optimal. F or order 19 we find all inequiv alen t saturated D-optimal designs with maximal determinant, 2 30 × 7 2 × 17, and co nfirm that the three known designs compris e a complete set. F or order 3 7 we prov e that the max ima l determinant is 2 39 × 3 36 , and find a sa mple o f inequiv alent saturated D-o ptimal designs. Our metho d is an extensio n of that used by Orr ic k to reso lve the previously smallest unknown o rder of 15; and by Cha djipantelis, Kounias and Moyssiadis to res olv e o rders 17 and 2 1 . The metho d is a t wo-step computation which first searches for candidate Gra m matrices and then attempts to decompo se them. Using a s imilar metho d, w e also find the complete sp ectrum of determinant v alues for { +1 , − 1 } matrices of order 13 . 1 In tro duction The Maximal Determinan t problem of Hadamard [12, 22] asks for the largest p ossible determinant of an n × n matrix w hose entries are dra wn from the set { +1 , − 1 } . W e are only interested in the absolute v alue of the determinan t, since we can alwa ys change the sign of the determinant b y c hanging th e sign of a row. The p roblem in its full generalit y has b een op en since firs t p osed by 1 Hadamard [12], and has applications to areas su c h as Exp erimen tal Design and Co ding Th eory . W e could equally w ell consider { 0 , 1 } matrices. There is a well -kn own mapping [25] from { 0 , 1 } ( n − 1) × ( n − 1) matrices to { +1 , − 1 } n × n matrices whic h m ultiplies the determinant b y ( − 2) n − 1 , and vice v ersa. T o a vo id confu sion w e only consider { +1 , − 1 } matrices. T heir determinan ts are alwa ys divisib le b y 2 n − 1 , thanks to the corresp ondence with { 0 , 1 } matrices. Th u s, it is con v enient to let D n denote max | det( R ) | , wh ere the maxim um is o ver all { +1 , − 1 } n × n matrices R , and d n = D n / 2 n − 1 . There is an extensiv e literature on the Maximal Determinan t problem, whic h splits into four cases, according to the v alue of n mo d 4. A general upp er boun d of D n ≤ n n/ 2 (1) on th e maximal determinan t applies to all the four c ases, but is not ac hiev- able u n less n = 1 , 2, or n ≡ 0 mo d 4. The conjecture th at this b ound is alw a ys ac hiev able when n ≡ 0 (mo d 4) is kno wn as the Hadamar d Conje c- tur e , and h as b een the sub ject of m uch inv estigation, see for example [11, 13]. Smaller u pp er b ounds are kn o wn for eac h of the other three equiv alence classes mod ulo f ou r . A b ound whic h h olds for all o dd orders, an d whic h is known to b e sharp for an infinite s equ ence o f ord ers congruent to 1 (mo d 4), is D n ≤ p ( n − 1) n − 1 (2 n − 1) , (2) due ind ep enden tly to Ehlic h [9] and Ba r ba [1]. A smaller up p er b ound , due to Ehlic h [10], applies only in the case n ≡ 3 (mo d 4): D n ≤ r ( n − 3) n − s ( n − 3 + 4 r ) u ( n + 1 + 4 r ) v  1 − ur n − 3+4 r − v ( r +1) n +1+4 r  . (3) Here s = 3 for n = 3 (and the factor ( n − 3) ( n − s ) is in terpreted as 1 in this case), s = 5 for n = 7, s = 6 for 11 ≤ n ≤ 59, s = 7 for n ≥ 63, r = ⌊ n/s ⌋ , v = n − rs , and u = s − v . The complicated form of the b ound (3) as compared with (2) is indicativ e of the extra difficulties whic h often seem to arise wh en n ≡ 3 (mo d 4). The b ound (3) is sharp when n = 3; it is not kno wn if it is sharp for any n > 3. In this w ork w e settle the smallest hitherto unr esolv ed case of n = 19. This case has remained op en desp ite higher orders (for example, 21) b eing solv ed b y similar metho ds, mainly b ecause the use of (2) and its generali- sation when the Gram matrix has a fixed blo c k— s ee Theorem 1—is muc h more effectiv e in pru ning the searc h tree than are (3) and its generalisatio n s. All orders congruen t to 3 m od 4 and larger than 19 are curr en tly open. In this pap er we only consider o dd ord ers. The smallest u nresolv ed orders which are congruent to 1 mo d 4 are n = 29 , 33 and 37. Of these, 2 w e resolv e n = 37, and impr ov e the up p er b ounds for n = 29 , 33. F or a summary , see T able 1 in § 7. Our metho d is structur ally similar to that u sed for n = 15 by Orrick [18], and b y earlier authors f or n = 17 in [17] and n = 21 in [4]. There are tw o essen tial steps, Gram fi nding and decomp osition. In th e cases w e consider, decomp osition b y hand would b e tedious for n = 19, and in feasible for larger orders. Th u s, we imp lemen t a bac k -tracking computer searc h to deal with this second step, describing suc h an algo rithm for the fir st time in the literature. Our Gram-fin d ing algorithm is discus sed in § 3, and our d ecomp osition algorithms in § 4. The results f or order 19 are describ ed in § 5, and for order 37 in § 6. In § 7 w e giv e some new upp er and lo wer b ounds for v arious o dd ord ers. F or orders n = 29, 33 , 45, 49 , 53 a n d 5 7 we ha ve not b een able to determine D n precisely , but we ha ve r educed the ga p b et ween th e kn o wn upp er and lo wer b ounds. Finally , in § 8 we a lso find the complete sp ectrum of determinant v alues for { +1 , − 1 } matrices of ord er 13. Previously , the sp ectrum w as only kno wn for orders up to 11. 2 Definitions Z denotes th e in tegers, and N 1 the p ositive intege rs. The follo w ing d efi - nitions are largely take n from [18], to whic h w e refer for further tec h nical definitions. Definition 1. A design is an m × n matr ix with entries dr awn fr om the set { +1 , − 1 } . If m = n the design is c al le d saturated . If the absolute value of the determinant of the satur ate d design is maximal for its or der, the design is c al le d D-optimal . In this pap er we consider saturated D-optimal d esig n s of o dd order. It is con v enient to consider “normalized” designs, leading to the next definition: Definition 2. A ve ctor with elements i n { +1 , − 1 } is parit y normalized i ff it has an even nu mb er of p ositive elements. A design is parity normalized iff al l its r ows and c olumns ar e p arity norma lize d. It is easy to sho w , as in [9, Lemmas 3.1, 3.2], that an y satur ate d d esign of o dd ord er can b e con v erted to a unique parit y normalized matrix by a series of negat ions of ro ws an d columns. If R 1 is a design, then any sig n ed p ermutatio n of the ro ws and columns of R 1 giv es another design R 2 , whic h we can regard as equiv alen t to the original d esign since | det( R 1 ) | = | det( R 2 ) | . W e can also change the signs of an y r o ws and /or columns of without changing more than the sign of the d eterminan t. This suggests th e follo win g defin itio n , in whic h a signed p erm utation matrix is a p ermutati on of the ro ws or columns of a d iag onal matrix diag( ± 1 , ± 1 , . . . , ± 1). 3 Definition 3. Two designs R and S ar e Hadamard equ iv alen t i ff S = P RQ for some p air of signe d p ermutation matric es ( P , Q ) . Definition 4. If R is a design, then G = RR T is c al le d the Gram matrix of R , and H = R T R is c al le d the d ual Gram matrix of R . Definition 5. Two symmetric matric es G 1 and G 2 ar e Gram equiv alen t iff G 1 = P G 2 P T for some signe d p ermutation matrix P . Definition 6. L et d min > 0 and let M n,p b e the set of squar e matric es M , of or der p , 1 ≤ p ≤ n , satisfying pr op erties 1–3 b elow. 1. M is symmetric and p ositive definite; 2. M i,i = n ; 3. M i,j ≡ n (mo d 4) . A matrix M ∈ M n,p is c al le d a candidate principal minor . If, further- mor e, n = p and the fol lowing additiona l pr op erties 4–5 hold: 4. d et( M ) = d 2 for d ∈ Z ; 5. d ≥ d min ; then M is c al le d a candidate Gram matrix . It is clear that Pr op erties 1, 2, 4 and 5 of candidate Gram matrices are satisfied b y all Gram matrices. F u rthermore, Prop ert y 3 of candid ate Gram matrices holds for Gram matrices G = RR T if R is assumed to b e parity normalized. 4 3 Gram-finding Algorithm W e summarize our Gram-find ing algorithm b elo w. The metho d is essentially that describ ed in greater detail in [18]. W e searc h for candidate Gram matrices wh ose determinan t is greater than or equal to a p ositiv e parameter d 2 min . 1. Set r = 1 and start from the candid ate principal minor M 1 = ( n ). 2. Incr ement r . Build a list of admissible vect ors f , and al lowable ve c- tors γ (for details see [18]). 3. F or eac h p ossible lexicographicall y maximal matrix M r − 1 of order r − 1, and eac h ad m issible v ector f , constru ct the matrix M r =  M r − 1 f f T n  . (4) If r = n , (a) if det( M r ) = d 2 ≥ d 2 min , output the ca nd idate G r am matrix M r . If r < n , (b) ev aluate d =     M r γ γ T 1     (5) for eac h allo wable ve ctor γ , looking for a “go od d ”, namely d suc h that the fun ctio n u r in Theorem 1 satisfies u r (1 , d ) ≥ d 2 min . If a go o d d is found, try to extend M r b y recursiv ely calling the algorithm (starting at step 2). Prunin g at step 3(b) of the ab o v e algorithm relies on the follo wing Theo- rem, originally used by Mo yssiadis and Kounias [17] to fin d a m aximal Gram matrix of order n = 17. Our v ersion b elo w con tains a sharp er b oun d (9) applicable when n ≡ 3 (mo d 4). 5 Theorem 1. [Enhanced Kounias & Mo yssiadis] L et M =  M r B B T A  b e a symmetric, p ositive definite matrix of or der n with elements taken fr om a set Φ whose memb ers ar e gr e ater than or e qual in magnitude to some numb er c , 0 < c ≤ n . H er e M r is a c andidate princip al minor of or der r ≤ n , and A is a squar e matrix of or der n − r , with diagonal elements A i,i = n . The c olumns of the r × ( n − r ) matrix B ar e taken fr om some set Γ r ⊆ Φ r . Define d ∗ and γ ∗ by d ∗ =     M r γ ∗ γ ∗ T c     = max γ ∈ Γ r     M r γ γ T c     . (6) Then det( M ) ≤ u r ( c, d ∗ ) , (7) wher e u r ( c, d ) = ( n − c ) n − r − 1 [( n − c ) det( M r ) + ( n − r ) max (0 , d )] . F urthermor e, if n ≡ 3 (mo d 4) , then the fol lowing b ounds apply: det( M ) ≤ ( n − 1) n − r det( M r ) + (8) [( n − 1) n − r − ( n − 3) n − r − ( n − r )( n − 3) n − r − 1 ] max(0 , d ∗ ) and, assuming det M r > ( n − 3) d et M r − 1 , det( M ) ≤ max k max b 1 ,...,b k ∈ N 1 b 1 + ... + b k = n − r max γ ∗ 1 ,...,γ ∗ k ∈ Γ r det      M r γ ∗ 1 j T b 1 · · · γ ∗ k j T b k j b 1 γ ∗ 1 T ( n − 3) I b 1 + 3 J b 1 · · · − J b 1 ,b k . . . . . . . . . . . . j b k γ ∗ k T − J b k ,b 1 · · · ( n − 3) I b k + 3 J b k      (9) wher e M r − 1 is the princip al ( r − 1) -by- ( r − 1) m inor of M r , j a is the c olumn ve ctor of dimension a whose elements al l e qual 1, J a,b is the a -by- b matrix whose elements al l e qual 1, and J a = J a,a . Pr o of. F or a p roof of inequ alities (7) and (8) w e refer to [18, Theorem 3.1 and Corollary 3.3]. A pro of o f (9) is sk etc hed in the App endix. The b ound (9) is sharp and th erefore p oten tially m u c h more p o werful than (7) or (8). Unfortunately , the m u ltidimen sional searc h for the optimal set of blo c k sizes ( b 1 , . . . , b k ), and t h e optimal set of vecto rs, { γ ∗ j } is exp en- siv e. W e therefore restricte d its u se to the situation w here the non-diagonal elemen ts of the last column of M r all equal − 1. This allo ws us to assume that all γ ∗ j consist entirely of elemen ts − 1, and w e are left only with th e searc h f or the optimal partition. Muc h of the computation associated with the latter searc h need only b e d one once. Th e u s e of (9) resulted in an appro ximately 15% improv ement in r unning ti me. 6 4 Decomp osition Algorithm The outp ut of the program describ ed in the previous section is a list L of candidate Gram matrices, co mp lete in the sens e th at it con tains one represent ativ e of eac h Gram equiv alence class with determinan t ≥ d 2 min for a give n b ound d min . W e n eed to d etermine if an y G ∈ L decomp oses as a pro du ct G = RR T for some square { +1 , − 1 } matrix R . Th is section describ es sev eral algo rith m s for carrying o u t this task. F or eac h candidate G this inv olv es a (p ossibly large) com b inato r ial searc h. It ma y b e regarded as searc hing a tree, where eac h lev el of the tree corre- sp onds to one ro w of the m atrix R . At lev el k w e kn o w k − 1 rows of R and try to fi nd a k -th ro w satisfying the constrain ts. Eac h no de at the k -th leve l corresp onds to one p ossible choice of the k -th ro w of R , giv en the pr eceding ro ws. If G = R R T has solutions, then the solution m atrice s, R , corresp ond to no des at lev el n of the tr ee. In p rinciple our p r ocedur e ma y generate man y Hadamard-equiv alen t solutions. W e p rune the tree so as to limit the n umb er of duplicate solutions pro duced. The searc h a lgorithm relie s on a family of constrain ts. The zeroth mem- b er of the family is a sp ecial case whic h can b e implemente d with a single Gram m atrix and w e call this constrain t the single-Gr am c onstr aint . The rest of the family require b oth the Gram matrix G = RR T and the dual Gram matrix H = R T R and w e call these Gr am-p air c onstr aints . Our decomp osition algorithm differs from that d escrib ed in [18] in s everal resp ects. First, it bu ilds up R by ro ws, instead of by rows and columns sim ultaneously . Second, it uses more general Gram-pair constraints (see (14) with j ≥ 2 b elo w). F or clarit y , we fir s t describ e a version of the algorithm which uses only the single-G r am constrain t. 4.1 Decomp osition using only t he single-Gram constrain t Denote the elemen ts of G b y g i,j for i, j = 1 , .., n , and the ro ws of R b y r i for i = 1 , . . . , n . T hen the constraint RR T = G is equiv alent to r i r T j = g i,j (10) for 1 ≤ i ≤ j ≤ n . The search tree is created b y application of these constrain ts. An outline of the basic a lgorithm is as fol lows. Th e main w ork is done by a recursive p rocedu re s earch ( k ) wh ic h searc hes an (implicit) subtree at lev el k , where the ro ot is at lev el 1. 7 Algorithm using the single-Gram constraint, v ersion 1 1. Initialize level k = 1, first r ow r 1 = (1 , 1 , . . . , 1) and R = r 1 . 2. Call searc h ( k ). 3. Ou tput “no solution” and halt. 4. sea rch ( k ): If k = n , outp ut the solution R and h alt . Otherwise in cre- men t k . Find all solutions r k ∈ { +1 , − 1 } n of the (un der-determined) set of sim ultaneous linear equations:      r 1 r 2 . . . r k − 1      r T k =      g 1 ,k g 2 ,k . . . g k − 1 ,k      F or eac h solution r k , app end r k to R and call se arch ( k ) recursiv ely to searc h the relev an t su btree. Retur n to the caller (i.e. backt rack). W e justify the c hoice of the fi rst row in the ab o v e algorithm by obs er v in g that G = R R T is in v arian t under a sig ned permutation of co lum ns of R , i.e. R 7→ R P for an y signed p erm utation matrix P . The ab o ve algorithm considers a large n umb er of equ iv alen t p artial so- lution matrices R , and is impractical for all b ut v ery small orders. W e can obtain a v astly more effici ent al gorithm by imp osing an ord ering constraint on the +1’s and − 1’s in row-v ectors. W e do this by defin ing the concept of “framings” and a n ew s et of asso ciated v ariables called “frame v ariables” whic h we use in Step 4 instead of r k . W e first d efine t h ese t erm s , and then giv e an impro ved v ers ion of the algorithm. A t eac h lev el k , we create a partition of the ind ices { 1 , . . . , n } into fr ames , where a frame is a nonempt y con tiguous set of ind ices; and the collec tion of frames is called a f r aming . A framing of size m is defin ed by a fr ame- widths ve ctor w = ( w 1 , . . . , w m ) ∈ N m 1 , P w i = n , where w i is th e size of the i -th fr ame. A t lev el 1 the framing consists of a single frame { 1 , . . . , n } with fr ame- w id th v ector w = ( n ). A t eac h subsequ en t level , the framing is a refinemen t of the framing from the p revious lev el. W e use the framing at lev el k − 1 to define the f rame v ariables that we use at lev el k in the follo w ing algorithm. The f rame v ariable x i giv es the n umb er of +1 en tries in the k -th ro w of R , co n sidering only the column indices giv en b y the i -th frame. Thus, the num b er of − 1 entries is w i − x i and the sum of the en tries is 2 x i − w i (see equation (1 1 )). 8 Algorithm using the single-Gram constraint, v ersion 2 1. Initialize the level k = 1, the fr ame-size m = 1 and the fr ame-width- ve ctor w = ( w 1 ) = ( n ). Set q 1 = (+1) and Q = q 1 . [In the course of the algorithm, q i is a column vecto r of size k − 1 or k , and Q is a matrix whose columns dep end on the q i . Also, w ma y b e though t of as a v ector of w eigh ts corresp ondin g to the columns of Q .] 2. Call searc h ( k ). 3. Ou tput “no solution” and halt. 4. sea rch ( k ): If k = n , outp ut th e solution Q and halt [here m = n ]. Otherwise incremen t k . Define in teger v ariables x 1 , x 2 , . . . , x m . Find all solutions to the fol- lo wing integ er p rogramming p roblem: Q      2 x 1 − w 1 2 x 2 − w 2 . . . 2 x m − w m      =      g 1 ,k g 2 ,k . . . g k − 1 ,k      (11) sub ject to 0 ≤ x i ≤ w i for 1 ≤ i ≤ m. (12) F or eac h solution, up date w and Q as follo ws: (a) Let w := ( x 1 , w 1 − x 1 , x 2 , w 2 − x 2 , . . . , x m , w m − x m ). (b) Recall that Q = ( q 1 , q 2 , ..., q m ) is a matrix of column v ectors q i , eac h of length k − 1. Up date Q to a k × 2 m matrix as follo w s: Q :=  q 1 q 1 q 2 q 2 · · · q m q m +1 − 1 +1 − 1 · · · +1 − 1  . (c) Compr ess Q b y removing all columns which corresp ond to zeros in w . (d) Compr ess w by remo ving all zero en tr ies. (e) Set m := length( w ). Call search ( k ) recursively to searc h the relev an t sub tree. When all solutions ha ve b een pro cessed, return to th e caller (i.e. bac ktrack). In p rocedu r e search ( k ) of v ersion 2 w e use Gaussian elimination with col- umn piv oting in order to find a ( k − 1) × ( k − 1) n onsingular minor of Q (this is alwa ys p ossible, since th e Gram matrix G is p ositive definite). W e then solve f or the corresp onding k − 1 b asic v ariables in terms of th e re- maining m − k + 1 non-b asic v ariables. The non-basic v ariables are c hosen 9 exhaustiv ely as int egers in the appr opriate interv als giv en by (12); the b asic v ariables are then determined un iquely (as real num b ers). I f the non-basic v ariables are not in Z or violate the b oun ds (12), the solution is discarded. It is preferable to c h oose as non-basic v ariables the v ariables w ith the small- est upp er b oun ds w i , pro vided that the resulting ( k − 1) × ( k − 1) minor is nonsingular. A heuristic f or acc omplish ing this is to weigh t the columns in prop ortion to the b ound s w i b efore p er f orming the Gaussian elimination. W e illustrate an iteration of the algorithm w ith an example. Consider the case n = 7 and the candidate Gram mat r ix (here and elsewhere w e may abbreviate “ − 1” b y “ − ”): G =           7 3 − − − − − 3 7 − − − − − − − 7 3 − − − − − 3 7 − − − − − − − 7 3 − − − − − 3 7 − − − − − − − 7           . Supp ose we are at lev el k = 3 in th e searc h . A t this stage the searc h tree has not branc hed y et, so there is ju st one matrix Q : Q =   1 1 1 1 1 1 − − 1 − 1 −   Asso ciate d w ith Q is the frame-widths v ector (at depth 3) whic h is w = (2 , 3 , 1 , 1) . W e comment that, translated in to the language of the algorithm in v ersion 1, Q and w together corresp ond to a 3 × 7 matrix R :   r 1 r 2 r 3   =   1 1 1 1 1 1 1 1 1 1 1 1 − − 1 1 − − − 1 −   . T o fin d the next row of Q , we define 4 (= m = | w | ) new v ariables x 1 , x 2 , x 3 , x 4 . The interpretatio n is that x 1 represent s the num b er of “+1”s in th e first w 1 = 2 en tries of ro w 4, x 2 represent s the num b er of “+1”s in the next w 2 = 3 en tries of row 4, as so on. W e u se the constrain ts imp osed by g 1 , 4 , g 2 , 4 , g 3 , 4 , giving the lin ea r system   1 1 1 1 1 1 − − 1 − 1 −       2 x 1 − 2 2 x 2 − 3 2 x 3 − 1 2 x 4 − 1     =   − − 3   . 10 The t wo in teger solutions whic h s atisfy th is system as well as the b ound s 0 ≤ x 1 ≤ 2, 0 ≤ x 2 ≤ 3, 0 ≤ x 3 ≤ 1, 0 ≤ x 4 ≤ 1 giv en b y equation (12) are ( x 1 , x 2 , x 3 , x 4 ) = (1 , 1 , 1 , 0) and (2 , 0 , 0 , 1). These generate t wo children in the searc h tr ee, with Q and w as follo ws: Q =     1 1 1 1 1 1 1 1 1 1 − − 1 1 − − 1 − 1 − 1 − 1 −     with w = (1 , 1 , 1 , 2 , 1 , 1); and Q =     1 1 1 1 1 1 − − 1 − 1 − 1 − − 1     with w = (2 , 3 , 1 , 1). The first Q lea ds to a solution; the second does not. 4.2 Decomp osition using Gram-pair c onstrain ts The al gorithm (v ersion 2) outlined in § 4.1, using only the s in gle- Gram con- strain t, quic kly b ecomes infeasible d ue to the s ize of the searc h s pace. It can b e imp ro v ed by noting th at, since the list L is complete, it must include a matrix H Gram-equiv alent to R T R . By p ermuting co lum ns of R , w e can assume that H = R T R . This r ela tion allo ws us to pr une the search more efficien tly than if w e did not kno w H . Recall that the c haracteristic p olynomial of a square matrix A is the monic p olynomial P ( λ ) = det( λI − A ). Sin ce H = R T G ( R T ) − 1 , the matrices G and H are simila r, so they hav e the same c haracteristic p olynomial. Th u s, the refined s trate gy is to consider eac h pair ( G, H ) ∈ L 2 , suc h that G and H hav e the same charac teristic p olynomial, and try to find R suc h that G = R R T , H = R T R . If w e h a ve considered ( G, H ) there is no need to consider ( H, G ) since this w ould corresp ond to the dual so lu tion R T . More precisely , consider the constrain t G j +1 = ( RR T ) j +1 = R ( R T R ) j R T = RH j R T . (13) W e say that the de gr e e of such a constr aint is j + 1, since th e elemen ts of G (not R ) o ccur with degree j + 1 . The case j = 0 corresp onds to the single-Gram constrain t consid ered in § 4.1. F or j = 1 we get the degree 2 constrain t G 2 = RH R T considered in [18]. The use of Gram-pair constrain ts with j > 1 is a new elemen t of our algorithm. In p rinciple, w e could get different constraints for j = 0 , 1 , . . . , n − 1. F or j ≥ n the Cayl ey-Hamilton theorem implies that w e get nothing new. 11 T o apply Gram-pair constraints for prun ing wh en only the firs t k − 1 ro ws of R are kno w n, partition th e matrices app earing in (13) into corresp ond ing blo c ks: R =  R 1 R 2  , G j +1 =  G 1 , 1 ( j + 1) G 1 , 2 ( j + 1) G 2 , 1 ( j + 1) G 2 , 2 ( j + 1)  , H j =  H 1 , 1 ( j ) H 1 , 2 ( j ) H 2 , 1 ( j ) H 2 , 2 ( j )  sa y , w here R 1 has k − 1 (kno wn) rows. Then w e can use the c onstr aints G 1 , 1 ( j + 1) = R 1 H 1 , 1 ( j ) R T 1 (14) since it only inv olves the kno wn rows of R . Th e matrices G j +1 and H j need only b e compu ted once. Observe th at H 1 , 1 ( j ) for j ≥ 2 dep ends on all the en tries in H . This suggests that (14) with j ≥ 2 ma y b e more effectiv e for prunin g than the “degree 2” case j = 1. In p ractic e, w e foun d that it was worth while to use (14) with both j = 1 and j = 2, bu t not with j > 2. When using Gram-pair constraints for pru n ing, we can no longer assume that the first ro w of R is (1 , 1 , . . . , 1). Th e algorithm (v ers ion 2) of § 4.1 has to b e m o dified s o step 1 starts with lev el k = 0, Q empty , and a frame-widths v ector w whic h is compatible with H , in the sense that H is in v arian t u n der p erm utations of ro ws (and corresp ondin g columns) within eac h frame. F or example, if w e tak e H = G in the example of order 7 ab o v e, w e can choose w = (2 , 2 , 2 , 1) as the initial frame-widths v ector. W e remark that w e used three v ariants of th e decomp ositio n algo rithm outlined in this subsection. O ne v arian t attempts to fi nd a decomp osition or (by failing to do so) to pro ve that a decomp osition of a giv en pair ( G, H ) do es not exist. A second v arian t fi nds all p ossible decomp ositions, up to Hadamard equiv alence. The outpu t t ypically includ es man y solutions that are Hadamard equiv alent, so we use McKa y’s program nauty [15] to remo ve all but one represent ativ e of eac h equiv alence class after transforming the problem to a graph isomorphism problem [14]. A th ir d v arian t is nondeter- ministic and attempts to tra verse the searc h tree b y c h oosing one or more c hildren randomly at eac h no de. This v arian t is usefu l in difficult cases where the deterministic v arian ts tak e to o long (see § 6. 2 for an example). 4.3 Pro ving indecomposabilit y using the H asse–M inko wski criterion A complemen tary approac h to the decomp osition problem, or, more prop - erly , t o proofs of indecomp osabilit y , mak es use of the Ha sse–Minko wski the- orem on rational equiv alence of qu adratic forms . The use of this theorem has a long history in d esign theory , originating with its use by Bruc k and Ryser in their p roof of their n onexiste n ce r esult f or certain finite pro jectiv e planes [3]. T amura recen tly applied the theorem to the question of decom- p osabilit y of candidate Gram mat r ices with block structure [27 ]. 12 Theorem 2. L et A and B b e symmetric, nonsingular r ational matric es of the same dimension. Then ther e exists a r ational matrix R such that B = RAR T if and only if 1. d et A / det B i s a r ational squar e , and 2. the p -signatur es of A and B agr e e for al l primes p and for p = − 1 . The criterion is implemen ted b y fi nding rational matrices U and V suc h that U AU T and V B V T are diagonal—this can alwa ys b e done—and th en by comparing the p -signatures of the resulting matrices for al l pr imes d ividing an y of the diago n al element s. T he p -signat u re of a diagonal form is defi n ed in [7, Chapter 1 5, § 5.1]. The application of this theorem is as follo ws: ther e is no decomp osition of the form R R T = G , R T R = H if there is a j ≥ 0 for wh ic h G j +1 fails to b e rationally equ iv alent to H j or for wh ich H j +1 fails to b e rationally equiv alent to G j . As was the case in the application of the Gram-pair constrain t in bac k-trac king searc h, w e need only c hec k the criterion for j < n . The Hasse–Mink o wski criterion is sometimes a comp etitiv e alternativ e to the bac ktrac king algorithm in ruling out the existence of a decomp osition. On certa in Gram m atrix pairs for whic h the bac k-trac kin g searc h ruled out a decomp osition only after exploring the searc h tree to great depth, the Hasse–Mink o wski criterion ruled out any d eco mp osition with a relativ ely fast compu tat ion. In most cases, how ever, bac k-trac king searc h is the m u c h faster metho d, esp ecially when r ational equiv alence fails only for large j , in wh ic h case the large-in teger arithmetic needed to implement the Hasse– Mink o wski criterion can b ecome prohibitiv ely exp ensiv e. F ur thermore, in a small num b er of cases, the Hasse–Mink o wski theorem fails entirel y to rule out a decomp osition w h ere bac ktrac kin g searc h succeeds. It is su rprising that this o ccurs relativ ely infrequentl y , as the existence of a decomp osition with R rational would app ear to b e a f ar milder constraint than the existence of a deco mp osition with R a { +1 , − 1 } matrix. 5 The Maximal Determinan t for Or d er 19 F or order 19, kno wn designs d ue to Smith [26], Cohn [6 ] and Orric k and Solomon [24] are D -optimal, as we n o w show. Theorem 3. The maximal determinant of { +1 , − 1 } or der 19 matric es is 2 30 × 7 2 × 17 = 833 × 4 6 × 2 18 . (15) Ther e ar e pr e cisely thr e e c orr e sp onding e quivalenc e classes of satur ate d D- optimal designs with r epr esentatives R 1 , R 2 and R 3 indic ate d in Figur e 2. Ther e ar e two c orr esp onding (Gr am e quivalenc e classes of ) Gr am matric es, G 1 = R 1 R T 1 = R T 1 R 1 and G 2 = R 2 R T 2 = R T 2 R 2 = R 3 R T 3 = R T 3 R 3 – se e Figur e 1. 13 Pr o of. A computational pro of of Theorem 3 is describ ed in §§ 5. 1 –5.2. Remark 1. The maximal determinant given by (15) is smal ler b y a factor 17 / √ 304 ≈ 0 . 975 than the Ehlich b ound (3) for n = 19 . 5.1 Candidate Gram-finding for or der 19 In the algorithm describ ed in § 3 w e u sed d min = 833 × 4 6 × 2 18 since { +1 , − 1 } matrices with this determinant w ere kno wn to exist. Our candidate Gram- finding program to ok 826 hours 1 to fin d nine equiv alence classes of candidate Gram matrices and to r ule out an y others. F or the nine candidate Gram matrices G , the v alues of p det( G ) / 2 30 w ere 840 (five times), 836 . 0625 (once), 836 (once), and 833 (t wice). The matrices are a v ailable from the w ebsite [2]. G 1 =                  n 3 3 3 3 3 3 − − − − − − − − − − − − 3 n 3 3 − − − − − − − − − − − − − − − 3 3 n 3 − − − − − − − − − − − − − − − 3 3 3 n − − − − − − − − − − − − − − − 3 − − − n 3 3 − − − − − − − − − − − − 3 − − − 3 n 3 − − − − − − − − − − − − 3 − − − 3 3 n − − − − − − − − − − − − − − − − − − − n 3 3 − − − − − − − − − − − − − − − − 3 n 3 − − − − − − − − − − − − − − − − 3 3 n − − − − − − − − − − − − − − − − − − − n 3 3 − − − − − − − − − − − − − − − − 3 n 3 − − − − − − − − − − − − − − − − 3 3 n − − − − − − − − − − − − − − − − − − − n 3 3 − − − − − − − − − − − − − − − − 3 n 3 − − − − − − − − − − − − − − − − 3 3 n − − − − − − − − − − − − − − − − − − − n 3 3 − − − − − − − − − − − − − − − − 3 n 3 − − − − − − − − − − − − − − − − 3 3 n                  G 2 =                  n − − 3 − − 3 − − 3 − − − − − − − − − − n 3 3 − − − − − − − − − − − − − − − − 3 n 3 − − − − − − − − − − − − − − − 3 3 3 n − − − − − − − − − − − − − − − − − − − n 3 3 − − − − − − − − − − − − − − − − 3 n 3 − − − − − − − − − − − − 3 − − − 3 3 n − − − − − − − − − − − − − − − − − − − n 3 3 − − − − − − − − − − − − − − − − 3 n 3 − − − − − − − − − 3 − − − − − − 3 3 n − − − − − − − − − − − − − − − − − − − n 3 3 − − − − − − − − − − − − − − − − 3 n 3 − − − − − − − − − − − − − − − − 3 3 n − − − − − − − − − − − − − − − − − − − n 3 3 − − − − − − − − − − − − − − − − 3 n 3 − − − − − − − − − − − − − − − − 3 3 n − − − − − − − − − − − − − − − − − − − n 3 3 − − − − − − − − − − − − − − − − 3 n 3 − − − − − − − − − − − − − − − − 3 3 n                  Figure 1: Optimal Gram matrices for n = 19. Here “ − ” stands for “ − 1” . 1 Computer times mentioned h ere and b elo w are for a single 2.3GHz Opteron pro cessor. In cases where the search could easily b e parallelised, we sometimes used sev eral pro cessors running in p aral lel. Our cand idate Gram-fin ding program actually took 188 hours using sever al pro cessors, each op erating on p art of the search tree. Our programs w ere written in C and used the GMP pack age to p erform multiple-precisio n arithmetic. 14 5.2 Decomp osition for order 19 Our d eco mp osition p rogram found that sev en of the candidate Gram matri- ces were indecomp osable, b u t the last tw o decomp osed (as exp ected). The runn in g time wa s only 0.85 sec. Nevertheless, it would b e extremely tedious to atte mp t to replicate the search by hand , since it inv olve s visiting ab out 1400 no des in the searc h trees. The nine matrices ha v e distinct c h aracteristic p olynomials, so we only had to consider the case G = RR T = H = R T R . Only the t w o candidate Gram matrices with smallest determinant w ere decomp osable, and th ese d e- comp osed in thr ee w ays, g iving thr ee Hadamard classes of designs (maxdet matrices) of order 19. S ee Figure 1 for the Gram matrices (note that they differ only in the fi rst r o w and column), and Figure 2 for t wo of the three designs. The third design R 3 can b e obtained from R 2 b y a switc h in g op er- ation, as indicated in Figure 2. A v ariant of our decomp osition program exhaustiv ely searc hes for all p ossible decomp ositions (up to equiv alence) of a giv en pair ( G, H ). Ru n- ning this program on ( G 1 , G 1 ) ga v e 11059 2 matrices in 36 seconds. Using McKa y’s program nauty [14, 15], w e v erified that they w ere all equiv alent to R 1 . Similarly , on ( G 2 , G 2 ) we obtained 3456 matrices in 3 seconds , and nauty v erified that 1728 were equiv alen t to R 2 , and the remaining 1728 were equiv alen t to R 3 . Thus, there are pr eci sely thr ee inequiv alent designs with maximal determinan t. 6 The Maximal Determinan t for Or d er 37 The case of order 37 w as h andled in muc h th e same wa y as order 19. Al- though 37 is m u ch larger than 19, w e h av e 37 ≡ 1 mo d 4, and t ypically the cases 1 m od 4 are easier than the cases 3 mo d 4 (as one can s ee from the summary at [22]). T his is partly b ecause Theorem 1 giv es a sharp er b ound when n ≡ 1 mo d 4. W e established that, for order 37, a kno wn design, foun d previously by Orric k and Solomon [21], is D-optimal. The design is not u nique, bu t the corresp onding Gram matrix is (up to equiv alence). Theorem 4. The maximal determinant of { +1 , − 1 } or der 37 matric es is 72 × 9 17 × 2 36 = 2 39 × 3 36 . (16) A r epr esentative R of one e quivalenc e class of satur ate d D- opt imal designs is indic ate d in Figur e 4. The c orr esp onding Gr am matrix is G = RR T = R T R as shown in Fig u r e 3. Mor e over, G is unique, up to e quivalenc e. 15 R 1 =                 + − − − − − − + + + + + + + + + + + + − + − − − − − − − − − + + − + + − + + − − + − − − − − − − + − + + − + + − + − − − + − − − − − − + + − + + − + + − − − − − + − − + + + + − − − + − − − + − − − − − + − + + + − + − − − + + − − − − − − − − + + + + − − + + − − − + − + + + + − − − − + + + − − − − + − + − + + + + − − − + − + − + − + − − − − + + + + + − − − + + − − − + − + − + − − + + − − − + + + − − + − − + + + − − − + − + − + − + − + − − + − + + + − − − + − − + + + − − − + − − + + + + − − − + + − − + + − − + − − − − + − − + + + + − + − − + + − − + − − − − + − + + + + − − + + − + + − − − − − − − + + + + + + − − + − + − − + + + + − − − + − − + − + − + + − + − − + + + − − − − + − + − − + − + + − + − + + + − − − − − +                 R 2 =                 − + + − + + − + + − − − − − − − − − − + − − − ⊕ ⊖ + − + + − − − ⊕ ⊖ + + − − + − − − ⊖ ⊕ + + − + − − − ⊖ ⊕ + − + − − − − + + + + + + + − − − + + − − − + + − + + − − − − + + + − − − − − − + + + + − + − − − + − + − + − − − − + − + − + + + − − + + + + − − + − − − + + − + − + + − + + − − − − + + + − − − − − + + − + + − + − − − + − + − + − − − − − + + + + + + − − + + + − − − + − − − − − − − ⊖ ⊕ − − + + + + + ⊖ ⊕ − + − − − − − − ⊕ ⊖ − + − + + + + ⊕ ⊖ − − + − − − − − − − + + + − + + + − − + − − + − − + + − − − + − − + − − + + + + − − − + − + − − − − + − − + − + + + − + − − + + − − − − − − + − − + + + + − − + − + − − − + + − − − + − − + − − + + + − − + − + − + − − − − + − − + − + + + − − − + + + − − − − − − + − − + + + +                 Figure 2: Two inequiv alen t saturated D-optimal d esigns of order 19. Here “ − ” sta n ds for “ − 1” and “+” stands for “+1”. A thir d inequiv alen t design R 3 is the same as R 2 except that the circled entries ha ve their signs reversed (this is an examp le of “switc hin g” , see Orric k [19]). 16 Remark 2. The maximal determinant given by (16) is smal ler b y a factor 8 / √ 73 ≈ 0 . 936 than the Ehlich-Barb a b ound (2) for n = 37 . G =           37 5 5 5 5 5 5 5 5 5 1 . .. 1 5 37 1 1 1 1 1 1 1 1 1 ... 1 5 1 37 1 1 1 1 1 1 1 1 ... 1 5 1 1 37 1 1 1 1 1 1 1 ... 1 5 1 1 1 37 1 1 1 1 1 1 ... 1 5 1 1 1 1 37 1 1 1 1 1 ... 1 5 1 1 1 1 1 37 1 1 1 1 ... 1 5 1 1 1 1 1 1 37 1 1 1 ... 1 5 1 1 1 1 1 1 1 37 1 1 ... 1 5 1 1 1 1 1 1 1 1 37 1 ... 1 1 1 1 1 1 1 1 1 1 1 37 ... 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 1 1 1 1 1 1 . .. 37           Figure 3: The optimal Gram matrix for n = 37. All omitted en tries are 1. The matrix R in Figure 4 wa s constructed b y Orric k and Solomon fr om a doubly 3- n ormaliz ed Hadamard matrix of order 36 . Th ere are at least 78 (and probably man y more) inequiv alen t d esigns, as we discu s s b elo w . 6.1 Candidate Gram matrices for order 37 Our bac ktrac king program with b ound d min = 2 39 3 36 (93.6% of the Ehlic h- Barba b ound) to ok 77 h ours to fin d 807 candidate Gram matrices. These had 284 distinct determinant s ∆ 2 in the range ∆ / (2 39 3 32 ) ∈ [81 , 85] . The candidate Gram matrices are a v ailable from the w ebsite [2]. 6.2 Decomp osition for order 37 W e app lied our decomp osition program to all pairs ( G, H ) of candidate Gram matrices where G and H had th e same c haracteristic p olynomial. There were 489 different c haracteristic p olynomials, and 1528 pairs ( G, H ) to consider. The decomp osition algorithm to ok 257 seconds to show th at 806 of the candidate Gr am matrices d id not decomp ose (in no case could more than t wo ro w s of R b e constructed). F or th e remaining candidate, w hic h w as in fact equiv alent to the Gram matrix G sh o wn in Figure 3, the program was stopp ed after ru nning for 147 hours and exploring about 1 . 7 × 10 8 no des (reac h ing lev el 26 of the tree ). A v arian t of ou r decomp osition program uses a r an d omised searc h – at eac h no de of the tree b eing searc hed, w e c ho ose to explore one (or sometimes t wo ) children selected uniformly at random. Using this randomised searc h program w e can d eco mp ose G , in fact we hav e no w f ou n d 39 solutions. Finding one sol u tion tak es on a verag e ab out 125 hour s . By a lso c onsid ering duals, w e get 78 solutions. Some (b u t not all) of th ese can b e obtained from a Hadamard m atrix of ord er 36, in the same wa y as the matrix R 17 of Figure 4. Since all 78 solutions are inequiv alent , w e exp ect that many more in equiv alen t solutions exist. Th e kno wn solutions are a v ailable from the w ebsite [2]. R =                                     + − − − − − − − − − + + + + + + + + + + + + + + + + + + + + + + + + + + + − − − − + + + + + + − − − − − − − − − + + + + + + + + + + + + + + + + + + − − − − + + + + + + + + + + + + + + + − − − − − − − − − + + + + + + + + + − − − − + + + + + + + + + + + + + + + + + + + + + + + + − − − − − − − − − − + + + + + − + − − + + − + + + + − − + − + + − + − + + + + + + − + + − − − + + + + + − − − + − + + + − + − + + + − − + + − + + + − − − + + + + + + − + + + + − + − + − + + + − + + − − + − + + − + + + + − + + − − + + + + − − + + + − + − + + − + − + − + − + + + + + − + + − + + − + + + + − − − + + − + + + − − + + − + + − + + + − + + − − + + − + + + − + − − + + + + + − + − + + + − − + − + + − + − + − + + + + + + + + − + − − + + + + − + − − + + + − + + − + − + + + + − + + − + − − + − + − + − + + − + + + − + + − + − − + − + + − + + − + + + − − − + + + − + − − + + − − + + + − − + − − + + + + + − + + − + + + + − − + + − + − − + + + − + − − + − + + + − − + + + − − + + − + + + − − + + + − − + + + + + − − + + − − + − − + + + − + − + + − + − + − + + + − − + + + − + − + + − − + + − − + + + + + − − − + + + − + − + − + − + + + − + + − + + + + − − − + + − − − − + + + − + + + + − − − − + + + + − + + + + + − − + + + + − + + − − − + + + + + − − − − − + + + + − − − + + − + + + + + − + − + + − + − − + + − + + − − − + + + − − − + + + − + + − + − + + + + + + − − − − − + − + + + + + + + − + − + − − + + − − − + + − + + + − + − + − + + + + − − + + + − + − + − + − + + − + − − + − − + − + + + + + − + − + + + − + − + + − + + − + − + + − + − + + − − + − + − − + + + − + + − + − + + + − + + + − + − − + − + − + + + + − − + − + − − + + + − + − + + − + + − + − + + − − + + + − + − + + + − + − + − + − − + − + − + + − + + + − + + − + + + − + − + − − + + + − + − + + − − + − + − + − + + + − + − + + − + + − + + + − + + − + + − − − + + − − + + − + − + + − + − + − + − + + + − + + + − − + + + + + + − − − + − − + + − − − + + + + + + − − + − − + + + − + + + − + + − − + + − − + + − + − + + − + + − − + − − + + − − + + + + + − + + + + − − + − − − − + + + + + − − − − + + + + + + + + + + − − − − + + + − + + − − + + + − + − − − + + + + − + + + + − − − + − + − + + + − − + + + − + − + + − + + − + + − + − − + + − + − − + + + − + − + + − − − + + + + + − + − − + + + − + − − + + + + − − + + + − − + + − + − − + + − + − + + + + − − + + + + − − + + + − + + − − − − − + + + + + − − + + − + + − − + + + + − − + + − + + − + + + + − + − − + − + − + − + − + + + − + − − + + − + + + − − − + + + + + + − − − + − + + + + − − + − − + + − + + + − + + − − + + + − + + − + − + + + − − + − + − + + + − − − + + − + − + − − + + − + + + + + − + + + − + − + − − + + + − + − − + − + + + − − + + − − + − + − + + + + + − + + + + − − − − + + + − − + + − + + + − − − + + − + + − + − + + −                                     Figure 4: A saturated D-optimal design of order 37, constru cted b y O rric k and Solomon. Here “ − ” stands for “ − 1” and “+” stands for “+1”. 7 Impro v ed Bound s for V ari ous Orders Recall that d n is th e maximal determinan t for order n , divided b y the kn own factor 2 n − 1 . T able 1 summ arises the b est known up p er and lo wer b ounds on d n for orders n = 19 , 29 , 33 , 37 , 45 , 49 , 53 , 57 (w e omit n = 25 , 41 , 61 b ecause for these orders the Ehlic h-Barba b ound (2) is attained). The figur es in paren theses are the ratios of the en tries to the Eh lic h-Barba b ound (for n ≡ 1 mo d 4) or the Ehlic h b ound (for n ≡ 3 mo d 4), round ed to thr ee decimals. F or n = 19 a nd n = 37, the u pp er an d lo wer b ound s are equal, and t hus optimal ( d n = u = ℓ in these cases). In the other cases the upp er b ounds are un atta inable, so d n ∈ [ ℓ, u ). In all cases the u pp er b oun d s are new, 18 and for n = 45 the low er b ound is new (the previous b est lo wer b ound w as 83 × 11 21 ). The last column giv es th e num b er of equiv alence classes of cand idate Gram matrices G with det( G ) ≥ (2 n − 1 u ) 2 . order n lo w er b ound ℓ upp er b ound u Gram coun t 19 833 × 4 6 (0 . 975 ) 833 × 4 6 (0 . 975 ) 9 29 320 × 7 12 (0 . 865 ) 329 × 7 12 (0 . 889 ) 9587 33 441 × 8 14 (0 . 855 ) 470 × 8 14 (0 . 911 ) 13670 37 8 × 9 18 (0 . 936 ) 8 × 9 18 (0 . 936 ) 807 45 89 × 11 21 (0 . 858 ) 99 × 11 21 (0 . 953 ) 1495 49 96 × 12 23 (0 . 812 ) 114 × 12 23 (0 . 965 ) 168 53 105 × 13 25 (0 . 788 ) 129 × 13 25 (0 . 968 ) 220 57 133 × 14 27 (0 . 894 ) 145 × 14 27 (0 . 974 ) 128 T able 1: Bound s on the (scaled) maximal determinan t d n 7.1 Discussion Let k = ⌊ n/ 4 ⌋ . F rom th e su mmary at [22] we observ e that, in the cases k ≤ 3 where d 4 k + 3 is kno wn precisely , d 4 k + 3 is divisible b y k 2 k − 1 . Ho we ver, our result f or n = 19 sh o ws that this pattern do es not con tin ue, f or d 19 is not divisible b y 4 7 . In all cases wher e d 4 k + 1 is known p recisely ( k ≤ 6 and k = 10 , 15 , 28 , . . . ), d 4 k + 1 is divisib le by k 2 k − 1 . T his is easily seen to b e true if the Ehlic h-Barba b ound (2 ) is attainable ( d 4 k + 1 is divisible by k 2 k in su c h cases), b ut it is also true for k = 2 , 4 , 5, where the Ehlic h -Barba b oun d is not atta inable. F or n = 29, we ruled out 9587 cand id ate Gram matrices to sho w that d 29 < 329 × 7 12 . If d 29 is divisible by 7 13 , then w e must ha v e d 29 = 322 × 7 12 = 46 × 7 13 , since this is the only multiple of 7 13 in the allo wable interv al [320 × 7 12 , 329 × 7 12 ). Ho we ver, all attempts to construct an example of order 29 with | det | / 2 28 > 320 × 7 12 , using hill-clim bin g or constructions based on Hadamard matrices of o r d er 2 8, h a ve failed. Thus, th e plausib le c onj ecture that d 4 k + 1 is divisible b y k 2 k − 1 ma y well b e false. An attempt to redu ce th e upp er b ound u to 322 × 7 12 is u nderw ay but ma y not b e f easible with our current r esources – so far w e hav e generated 16683 candidate Gram matrices (taking ab out t w o pro cessor-y ears) but estimate that there are ab out 2200 00 in all. F or n = 33 w e ruled out 1367 0 candidate Gram matrices to establish an upp er b ound of u = 470 × 8 14 . It is un lik ely th at we can r educe u m u c h further without impro vemen ts in the candidate Gram-fin ding p r ogram, since it to ok ab out t wo pro cessor-y ears to generate th e candidates for u = 470, though only about six hours to sho w that none of th em decompose. 19 F or n = 45, the n ew low er b ound of 89 × 11 21 w as established by a construction usin g a doubly 3-normalized Hadamard matrix of order 44. Details will app ear elsewher e. 8 The Sp ectrum for Order 13 The sp e ctrum S n of the determinant function for { +1 , − 1 } m atric es is de- fined to b e the set of v alues tak en b y | det( R n ) | / 2 n − 1 as R n ranges ov er all n × n { +1 , − 1 } matrices. F or 2 ≤ n ≤ 7, the sp ectrum includes all in tegers b et w een 0 and d n . The sp ectrum for n = 8 was fir st compu ted b y Metrop olis, Stein, and W ells [16], who f ound th at gaps o ccur, in fact S 8 = { 0 , 1 , . . . , 18 , 20 , 24 , 32 } . A (non-computer-based) p ro of of th e exis- tence of gaps w as later giv en by Craigen [8 ]. A t present, th e sp ectrum is kno wn for n ≤ 11 and (giv en here for the first time) for n = 13. The results for n = 9 are du e to ˇ Zivk o vi´ c [28] (and, indep endently , Charalam bides [5]), those for n = 10 are due to ˇ Zivk o vi´ c [28], and those for n = 11 are due to Orric k [18]. The sp ectra for n ≤ 11, and conjectured s p ect r a for n = 12 and 14 ≤ n ≤ 17, m ay b e found at [23]. Here w e giv e only the sp ectrum for n = 13, using the notatio n a..b as a shorthand to repr esent the int erv al { x ∈ Z : a ≤ x ≤ b } . Theorem 5. The sp e ctrum for or der 13 is S 13 = { 0 .. 2172 , 2174 .. 2185 , 2187 .. 2196 , 2199 .. 2202 , 2205 , 220 8 , 2210 , 2211 , 2214 .. 2218 , 222 0 .. 2226 , 222 8 , 22 29 , 2 230 , 2232 , 2233 , 2235 , 2238 , 2240 , 2241 , 2243 .. 2245 , 224 7 , 22 48 , 2250 , 2253 , 2 256 , 2258 .. 2260 , 2262 , 226 4 , 2265 , 2267 , 2268 , 227 1 , 227 2 , 2274 , 2277 , 2280 , 2283 , 2286 , 2288 , 2292 , 2295 , 2296 , 2304 , 2307 , 231 2 , 231 3 , 2316 , 2319 , 2320 , 2322 , 2325 , 2328 , 2331 , 2334 , 2336 , 2340 , 2343 , 234 4 , 234 9 , 2352 , 2355 , 2360 , 2361 , 2367 , 2368 , 2370 , 2373 , 2376 , 2385 , 2394 , 240 0 , 240 3 , 2406 , 2421 , 2430 , 2432 , 2439 , 2457 , 2472 , 2484 , 2496 , 2511 , 2520 , 253 8 , 256 0 , 2583 , 2592 , 2619 , 2646 , 2673 , 2835 , 2916 , 3159 , 3645 } . Pr o of. The pro of is computational. Using a heuristic algorithm describ ed in [20, pg. 34] , w e found examples of o r d er 1 3 matrices with all 2173 deter- minan ts 0 , 1 × 2 12 , 2 × 2 12 , . . . , 2172 × 2 12 . T he fir s t “gap” was at 2173 × 2 12 . W e ran the Gram-finding p rogram of § 3 with lo w er b oun d d min = 2173 × 2 12 . It p rod uced 8321 ca nd idate Gram matrices in 73 min utes. W e then r an the decomp osition program of § 4 whic h f ou n d (in 48 seconds) that 1643 of the candidate Gram matrices decomp osed, giving 130 distinct determinan ts in the range [2174 , 3645 ]. These are listed in th e statemen t of the theorem. App endix: Pro of of new b ound (9) in Theorem 1 The p roof is a generalisation Ehlic h’s p ro of [9] of the b ound (3) on the maximal determinant, w h ic h applies in the case n ≡ 3 (mo d 4) . Eh lic h’s 20 pro of sho ws the foll owing. (1) The candid ate pr incipal min or of m aximal d etermin ant has non-diagonal elemen ts equal to either − 1 or 3. (2) It is a blo ck matrix , whic h means th at the non-diagonal 3s o ccur in square blo c ks along the diagonal. (3) In the case that the candidate pr incipal min or is a candidate Gram matrix, that is, its size is n , th e num b er of blo c k s is s , with u blo c k s of size ⌊ n/s ⌋ and v blo c ks of size ⌊ n/s ⌋ + 1, where s , u , and v are defi n ed follo w ing (3). W e generalise this to the case where the ca nd idate principal minor contai n s a fixed pr incipal s u bmatrix M r . If su c h a candidate p rincipal minor of maximal determinan t is written as  M r B B T A  , then our result, assuming the hyp othesis det M r > ( n − 3) det M r − 1 , is that A satisfies prop erties (1) and (2) ab ov e, and the su b matrices of B corresp onding to blo c ks of A consist of rep eated columns. The generalisation of (3) dep ends on M r and is not unique in general. F rom no w on, w e tak e n ≡ 3 (mo d 4) and n > 3. Define C m = { C m | C m = ( c ij ) , c ij = c j i , C m is p os. def. , c ii = n, c ij ≡ n (mo d 4) , i, j = 1 . . . m } . (17) and E m = { E m | E m ∈ C m and the lea din g r × r subm atrix of E m is M r } . (18) Define C ∗ m and E ∗ m (whic h may not b e unique) b y the conditions det C ∗ m = max { det C m | C m ∈ C m } , det E ∗ m = max { det E m | E m ∈ E m } . The first thing Ehlic h p ro ves (Th eorem 2.1) is that det C ∗ m > ( n − 3) det C ∗ m − 1 for 2 ≤ m ≤ n . T o explain the use of this theorem, w e first in tro duce a notation. If C m ∈ C m then d efine ˜ C m b e the matrix that results from replacing the last diag onal elemen t of C m b y 3, i.e. ˜ C m = ( ˜ c ij ) where ˜ c ij = ( 3 if i = j = m c ij otherwise. 21 Expanding det C ∗ m b y min ors on its last ro w, w e find det C ∗ m − det ˜ C ∗ m = ( n − 3) d et C m − 1 ≤ ( n − 3) d et C ∗ m − 1 (19) where C m − 1 is the lea d ing ( m − 1) × ( m − 1) submatrix of C ∗ m . (Note that ˜ C ∗ m is the r esult of applying the tilde op eratio n to C ∗ m .) The theorem th en implies that det ˜ C ∗ m > 0 and therefore that ˜ C ∗ m is p ositive defin ite . This b ecomes imp ortan t in later pr o ofs when ev aluating determinants that a rise as the result of column op erations. F or the generalization to our case, w e app ear to need the extra condition det ˜ M r > 0. This cannot b e exp ected to h old in general. Consider for example, M 2 =  11 7 7 1 1  (20) for which det ˜ M 2 < 0. F or n o w , w e lea v e it as a question for empirical study wh ether the condition h olds often enough in practical searc hes for th e follo w ing considerations to b e us efu l. Theorem 6. L et 1 ≤ r ≤ m , let M r ∈ C r , and let det ˜ M r > 0 . Then the set of elements E m ∈ E m for which det ˜ E m > 0 is non-empty. Pr o of. W e prov e the theorem b y ind uction on m . F or the base ca se, m = r , the m atrix M r ∈ E r satisfies d et ˜ M r > 0 b y assumption. No w assu me that the theorem holds f or all E k with r ≤ k ≤ m . W e wan t to show that it holds for E m +1 . Let E m b e an elemen t of E m for whic h det ˜ E m > 0, and write this elemen t as E m =  C m − 1 γ γ T n  . (21) F orm the matrix E m +1 =   C m − 1 γ γ γ T n 3 γ T 3 n   . (22) Subtracting column m fr om column m + 1 and exp anding b y minors on column m + 1 we fi nd that det E m +1 = ( n − 3) det E m + ( n − 3 ) det ˜ E m . Both terms are p ositiv e, wh ic h means that d et E m +1 is p ositive , and therefore that E m +1 is p ositiv e definite and h ence an elemen t of E m +1 . By a s imilar computation det ˜ E m +1 = ( n − 3) det ˜ E m > 0. Th erefore E m +1 is a su ita b le elemen t. Theorem 7. L et 1 ≤ r < m , let M r ∈ C r , and let d et ˜ M r > 0 . Then det E ∗ m > ( n − 3) d et E ∗ m − 1 . 22 Pr o of. When m = r + 1 w e write M r =  C r − 1 γ γ T n  (23) and define E r +1 =   C r − 1 γ γ γ T n 3 γ T 3 n   . (24) F rom the pro of of th e previous theorem w e kno w that E r +1 ∈ E r +1 and det ˜ E r +1 > 0. No w det E ∗ r +1 ≥ det E r +1 = ( n − 3) det M r + ( n − 3) det ˜ M r > ( n − 3) det M r = ( n − 3) d et E ∗ r . W e no w pro ceed by induction. Assume t h at det E ∗ m > ( n − 3) det E ∗ m − 1 . W rite E ∗ m =  E m − 1 γ γ T n  (25) and define E m +1 =   E m − 1 γ γ γ T n 3 γ T 3 n   . (26 ) No w d et E m +1 = ( n − 3) det E ∗ m + ( n − 3) det ˜ E ∗ m . Note th at det E ∗ m = ( n − 3) det E m − 1 + det ˜ E ∗ m ≤ ( n − 3) det E ∗ m − 1 + det ˜ E ∗ m . F rom the indu ctio n h yp othesis, it follo ws that det ˜ E ∗ m > 0, and so det E m +1 > ( n − 3) det E ∗ m . This pro of conta ins the pro of of an imp ortan t corollary: Corollary 8. L et 1 ≤ r ≤ m , let M r ∈ C r , and let det ˜ M r > 0 . Then det ˜ E ∗ m > 0 . W e n o w generalize Eh lich’s Theorem 2.2. Again we need the assu mption that det ˜ M r > 0. Theorem 9. L et 1 ≤ r ≤ m , let M r ∈ C r , and let d et ˜ M r > 0 . Write E ∗ m =  M r B B T A  , wher e A = ( a ij ) and B = ( b ij ) satisfy the c onditions in the definition of E m . Then for i 6 = j we have a ij = − 1 or 3 . Pr o of. When m = r or r + 1 the s tat ement is v acuously true. S o we assu me m ≥ r + 2. Su pp ose ther e is an elemen t a ij = c 6 = − 1 or 3 for i 6 = j . Th en | c | > 3. W e ma y assume that th e element is p ositioned so that i = m − 1, j = m , so that w e may write E ∗ m =   E m − 2 α β α T n c β T c n   . 23 By interc h an ging the last t wo rows and last t w o columns, if necessary , w e ma y assume that det  E m − 2 α α T n  ≤ det  E m − 2 β β T n  . W e no w claim that the matrix E m =   E m − 2 β β β T n 3 β T 3 n   has larger determinant than E ∗ m , a cont rad iction. T o establish the claim, ev aluate b oth determinan ts: det E ∗ m = ( n − 3) d et  E m − 2 α α T n  + det ˜ E ∗ m det E m = ( n − 3) d et  E m − 2 β β T n  + det ˜ E m By Corollary 8, ˜ E ∗ m is p ositiv e d efinite. Symmetric row and column op era- tions do not affe ct p ositiv e d efiniteness, s o w e get det ˜ E ∗ m = det    E m − 2 α β α T n c β T c 3    = det    E m − 2 β α − c 3 β β T 3 0 α T − c 3 β T 0 n − c 2 3    ≤  n − c 2 3  det " E m − 2 β β T 3 # . W e ev aluate d et ˜ E m b y su btracting column m fr om column m − 1 and doing expansion b y minors on col u mn m − 1 to obtain det ˜ E m = ( n − 3) d et  E m − 2 β β T 3  . Since | c | > 3 w e ha ve det ˜ E ∗ m < det ˜ E m and therefore det E ∗ m < det E m . No w w e w an t to generalize Ehlic h’s Theorem 2.3. First a usefu l lemma ab out blo c k matrices. (See Ehlic h’s pap er f or the formal d efi nition of blo c k.) Lemma 10. A symmetric matrix A = ( a ij ) with diagonal elements n is a blo ck matrix if a nd only if its no n- diago nal elements ar e al l − 1 or 3 and for any i 6 = j such that a ij = 3 the c olumns i and j differ only in their i th and j th elements. 24 Pr o of. If A is a b lo c k matrix, the statemen t is clearly tru e. F or the conv erse, let ( i, j ) b e t h e position of one of the 3s of A . Define i 1 = j , i 2 = i , and let { i 2 , i 3 , . . . , i p } b e the set of all indices h for which a hj = 3. Let 2 ≤ k ≤ p . Then a i k j = 3 means that the i th k elemen t of column j is 3. Let 1 ≤ ℓ ≤ p , ℓ 6 = k . Th en sin ce column s i ℓ and j agree in their i th k elemen t w e hav e a i k i ℓ = 3 for all k 6 = ℓ . F or an index h / ∈ { i 1 , . . . , i p } w e ha ve a hj = − 1. But since columns j and i ℓ , 1 ≤ ℓ ≤ p , agree in their h th elemen t, we hav e a hi ℓ = − 1 for all 1 ≤ ℓ ≤ p . Therefore the set of indices { i 1 , . . . , i p } forms a b lock. Hence ev ery one of the 3s in A lies in a blo c k, and A is a bloc k matrix. No w for the generaliza tion of Ehlic h’s Theorem 2.3. Theorem 11. L e t 1 ≤ r ≤ m , let M r ∈ C r , and let d et ˜ M r > 0 . Write E ∗ m =  M r B B T A  , wher e A = ( a ij ) and B = ( b ij ) satisfy the c onditions in the definition of E m . If for som e i 6 = j , a ij = 3 , then c olumns i and j of B ar e e q ual and c olumns i and j of A ar e e qual exc ept f or their i th and j th elements. Pr o of. By Theorem 9 w e kno w that the non-diagonal elemen ts of A are − 1 or 3. As b efore, the theorem is v acuously tru e if m = r or m = r + 1. Assume m ≥ r + 2 and le t A ha ve an elemen t 3, sa y in p ositi on ( m − 1 , m ). W rite E ∗ m =     M r B 1 β m − 1 β m B T 1 A 1 α m − 1 α m β T m − 1 α T m − 1 n 3 β T m α T m 3 n     . W e assu me, as we ma y (b y sw appin g the last tw o rows and last t wo columns if necessary), that det   M r B 1 β m − 1 B T 1 A 1 α m − 1 β T m − 1 α T m − 1 n   ≤ det   M r B 1 β m B T 1 A 1 α m β T m α T m n   . Our goal is no w to sho w that β m − 1 = β m and α m − 1 = α m . S upp ose that this is not th e ca se. W e claim that det E m > det E ∗ m where E m =     M r B 1 β m β m B T 1 A 1 α m α m β T m α T m n 3 β T m α T m 3 n     . W rite det E ∗ m = ( n − 3) d et   M r B 1 β m − 1 B T 1 A 1 α m − 1 β T m − 1 α T m − 1 n   + det ˜ E ∗ m 25 and det E m = ( n − 3) d et   M r B 1 β m B T 1 A 1 α m β T m α T m n   + det ˜ E m . The first term on the righ t in det E ∗ m is no larger than the first term on the righ t in det E m , and we will see that the second term of det E ∗ m is strictly smaller than the second term of det E m . By sub tr ac ting row and column m of ˜ E m from ro w and column m − 1 of ˜ E m and expanding by minors on column m − 1 w e find that det ˜ E m = ( n − 3) d et   M r B 1 β m B T 1 A 1 α m β T m α T m 3   . On the other hand, using Corollary 8 , whic h imp lies that ˜ E ∗ m is p ositive definite, and ev aluating th e determinan t as w e did for det ˜ E m , w e find that det ˜ E ∗ m < ( n − 3) d et   M r B 1 β m B T 1 A 1 α m β T m α T m 3   . This f ollo ws b ecause α m − 1 − α m and β m − 1 − β m , which together form th e first m − 2 elements of column m − 1 in the expansion by minors, are not b oth zero. Corollary 12. The matrix A in The or em 11 is a blo c k matrix. Pr o of. W e hav e prov ed in Theorems 9 and 11 that b oth of the conditions needed in Lemma 10 for A to b e a blo c k matrix hold. Our conclusion is th at, wh en det ˜ M r > 0, the maximal d eterminan t completion E ∗ m of M r tak es a form where A is a blo c k matrix with some n umb er k of blo c ks wh ose sizes we will denote b 1 , b 2 , . . . , b k , and w here B =  B 1 . . . B k  with eac h of the matrices B j a rank-1 matrix consisting of a co lumn β ∗ j rep eated b j times. Th is establishes (9). Ac kno wledgemen ts W e gratefully ackno wledge sup p ort from the Australian Research Council, the ARC Cen tre of Excellence f or Mathematics and Statistics of C omplex Systems (MASCO S), and INRIA via the ANU-INRIA Asso ciate T eam ANC. 26 References [1] G. Barba, In torno al teorema di Hadamard sui determinanti a v alore massimo Giorn . M at. Battaglini 71 (1933 ), 70–86 . [2] R. P . Brent , W. P . Orric k, J. H. Os b orn and P . Zimmermann, Some D- optimal designs of ord er s 19 and 37, http :// www maths.anu.edu.au/ ~ brent/maxdet / [3] R. H. Bruck and H. J. Ryser, T he nonexistence of certain finite p ro jec- tiv e planes Canadian J. Math. 1 (194 9) 88–93 . [4] T. Chadj ipan telis, S. Kounias and C . Mo yssiadis, The m axim um deter- minan t of 21 × 21 (+1 , − 1)-matrices and D- optimal designs, J. Statist. Plann. Infer enc e 16 , 2 (19 87), 167–178. [5] A. Charalam bides, p ersonal co mmunication to W. Orric k, 2005. [6] J. H. E . Cohn , Almost D-optimal d esigns, Utilitas Math. 57 (2000), 121–1 28. [7] N. J. A. Sloane and J . H. Conw a y , Spher e P ackings, L attic es and Gr oups, volume 290 of Grund lehr en der Mathematischen Wis- senschaften , Springer, New Y ork–B erlin–Heidelb erg, 3rd edition, 1998. [8] R. Craige n , The range of the determinant function on the set of n × n (0 , 1)-matrices, J. Combin. Math. Combin. Comput. 8 (1990), 161–171. [9] H. Ehlich, Determinan tenabsch¨ atzungen f ¨ ur bin¨ are Matrizen, Math. Z. 83 (196 4), 123– 132. [10] H. Ehlic h, Determinan tenabsch¨ atzungen f ¨ ur bin ¨ are Matrizen mit n ≡ 3 mo d 4, Math. Z. 84 (1964 ), 438–4 47. [11] A. V. Geramita and J. Seb erry , Ortho gonal Designs: Q u ad r atic F orms and Hadamar d M atr ic es , Marcel Dekk er, New Y ork, 1979. [12] J. Ha d amard, R ´ esolution d’une question r elat ive aux d´ eterminants, Bul l. des Sci. Math. 17 (18 93), 240–246. [13] K. J. Horadam, Hadamar d Matric es and their Applic ations , Princeton Univ ersity P r ess, 200 7. [14] B. D. McKa y , Hadamard equiv alence via graph isomorphism, Discr ete Mathematics 27 (1979), 213–2 14. [15] B. D. McKa y , nauty , http :// cs. anu. ed u.au/ ~ bdm/nauty/ . 27 [16] N. Metropolis, Sp ectra of d etermin ant v alues in (0 , 1) matrices. In A. O. L. Atkin and B. J. Birc h, editors, Computers in Numb er The ory: Pr o c e e dings of the Scienc e R ese ar ch Atlas Symp osium No. 2 held at Oxfor d, 18–23 August, 1969 , Academic Press, London, 1971 , 271 –276. [17] C. Mo yssiadis and S . Kounias, T he exact D-optimal first order s atu- rated d esign with 17 observ ations, J. Statist. Plann. Infer enc e 7 (1982), 13–27 . [18] W. P . Or ric k, The maximal {− 1 , 1 } -determinan t of order 15, M e trika 62 (200 5), 195– 219. [19] W. P . Orrick, Switc hing op erations for Hadamard matrices, SIAM J. Discr ete Math. 22 (20 08), 31–50. [20] W. P . Orric k, Range and d istribution of determinant s of binary matri- ces, invited talk present ed at the Maximal Determinant Workshop held at the Au stralian National Unive rs ity , 17 Ma y 2010. Av ailable from http://mypage.iu.edu/ ~ worrick/talks.html . [21] W. P . Orr ic k and B. Solomon, Large-determinant sign matrices of order 4 k + 1, Discr ete Math. 307 (2007) , 226–23 6. [22] W. P . Or ric k and B. Solomon, The Hadamar d maximal determinant pr oblem , http:/ / www.india na. edu / ~ maxdet/ . [23] W. P . Orric k and B. Solomon, Sp e ctrum of the determinant function , http://www.indiana.edu/ ~ maxdet/spectrum . html . [24] W. P . O rric k and B. Solomon, Conje ctur e d 19 × 19 {− 1 , +1 } matri- c es of maxima l determinant , 15 May 2003. http: // www.ind iana.edu/ ~ maxdet/d19.html . [25] J. H. Osb orn, The Hadamar d Maximal Determinant Pr oblem , Hon- ours thesis, Un iv ers ity of Melb ourne, 2002. Av ailable from ht tp:// wwwmaths . a nu.edu.au/ ~ osborn/publicat ions/ pubsa ll. html . [26] W. D. S m ith, Studies in Computatio nal Ge ometry Motivate d by Mesh Gener ation , PhD dissertation, Princeton Univ ersit y , 1988. [27] H. T am u ra, D-optimal designs and g rou p divisible designs, J. Combin. Des. 14 (2006), 451– 462. [28] M. ˇ Zivk o vi´ c, C lassificat ion of sm all (0 , 1) matrices, Line ar Algebr a Appl. 414 (200 6), 1, 31 0–346. 28