Generating binary partial Hadamard matrices

Gene a ing bina y pa ial Hadama d ma ices V. Ál a ez a, J.A. A ma io a, R.M. Falcón a, M.D. F au a, F. Gudiel a, M.B. Güemes b, A. Osuna a a Dp o. Ma emá ica Aplicada I, Uni e sidad de Se illa, Spain b Dp o. Álgeb a, Uni e sidad de Se illa, Spain Keywo ds: Pa ial Hadama d ma ix Hadama d g aph Clique Cons ain sa is ac ion p oblem a b s a c This pape deals wi h pa ial bina y Hadama d ma ices. Al hough he e is a as simple way o gene a e abou a hal (which is he bes asymp o ic bound known so a , see de Launey (2000) and de Launey and Go don (2001)) o a ull Hadama d ma ix, i canno p o ide la ge pa ial Hadama d ma ices beyond his bound. In o de o o e come such a limi a ion, we in oduce a pa icula subg aph G o I o’s Hadama d G aph ∆(4 ) (I o, 1985), and s udy some o i s p ope ies, which acili a es ha a p ocedu e may be designed o cons uc ing la ge pa ial Hadama d ma ices. The key idea is ansla ing he p oblem o ex ending a gi en clique in G in o a Cons ain Sa is ac ion P oblem, o be sol ed by Minion (Gen e al., 2006). Ac ually, i e a ion o his p ocess ends wi h la ge pa ial Hadama d ma ices, usually beyond he bound o hal a ull Hadama d ma ix, a leas as ou compu a ion capabili ies ha e led us hus a . 1. In oduc ion Bina y Hadama d ma ices consis in {1,−1}-squa e n×nma ices Hsuch ha HHT=nIn. I may be s aigh o wa dly checked ha such a ma ix Hmus be o size 1, 2 o a mul iple o 4, as soon as h ee ows a e assumed o be mu ually o hogonal. Ac ually, i seems ha Hadama d ma ices migh exis o e e y o de mul iple o 4, as he Hadama d Conjec u e claims, al hough his ac emains s ill open o mo e han a cen u y (see [10] o u he de ails). F om a p ac ical poin o iew, aking in o accoun possible applica ions, some imes he e is no need o conside a ull Hadama d ma ix. In ac , i su ices o mee a la ge amoun o pai wise o hogonal ows. This has o igina ed he in e es in cons uc ing bina y pa ial Hadama d ma ices, ha is, m×4 (1,−1)-ma ices PH sa is ying PH ·PHT=4 Im, o m≤4 . We call m he dep h o PH. Fo ins ance, exis ence o la ge pa ial Hadama d ma ices implies la ge lowe bounds on he size o se e al in e es ing combina o ial objec s, as poin ed ou in [5]. Un o una ely i seems ha hei explici cons uc ion is equally ha d as well. De Launey p o ed in [4] ha pa ial Hadama d ma ices o size abou a hi d o a ull 4 ×4 Hadama d ma ix exis o la ge . The p oo gi es a polynomial ime algo i hm in o cons uc ing such a ma ix. Fu he mo e, De Launey and Go don p o ed in [5] ha abou a hal o a Hadama d ma ix 4 ×4 exis s o la ge , assuming ha he Riemann hypo hesis is ue. The idea was decomposing 2 −ias he sum o iodd p ime numbe s pi, 2 ≤i≤3, so ha he jux aposi ion o he co esponding Paley con e ence ma ices p o ides a pa ial Hadama d ma ix o dep h 2 min{pi} + 2. F om a heo e ical poin o iew, one could p oceed e en in a simple way, a oiding his assump ion conce ning he Riemann hypo hesis, which is jus used in o de o p o ide an explici cons uc i e algo i hm wo king in polynomial ime. Ac ually, gi en any posi i e in ege , conside he se PH o pai s o in ege s ( 1, 2) such ha 1≤ ⌊ 2⌋ ≤ 2, 1+ 2= and some ull Hadama d ma ices H4 1and H4 2exis o o de s 4 1and 4 2, espec i ely. Le m=max( 1, 2)∈PH 1. Then a pa ial Hadama d ma ix PH o o de 4m×4 may be s aigh o wa dly cons uc ed, as soon as any ull Hadama d ma ix o o de 4mand a collec ion o wha e e 4m ows o any ull Hadama d ma ix o o de 4( −m) a e jux aposed. No ice ha , as de ined, 4mis necessa ily close o 2 , since i may a ely occu ha sys ema ically no ull Hadama d ma ices exis o o de s 4 1and 4 2, o 1≤ 2, 1∼ 2, 1+ 2=2 , no ma e he Hadama d Conjec u e has no been p o ed ye . As a ma e o ac , a look a he upda ed lis o in ege s <500 o which no Hadama d ma ices o o de 4 a e known (namely, 167,179,223,283,311,347,359,419,443,479,487,491 [7]), suppo s his idea. Un o una ely, none o hese me hods can p o ide a pa ial Hadama d ma ix o dep h g ea e han hal o a ull Hadama d ma ix. The aim o his wo k is o desc ibe an al e na i e p ocedu e p o iding la ge pa ial Hadama d ma ices as well, hope ully beyond his bound. Ac ually, his will be he case o he examples we ha e wo ked ou . The wo k may be summa ized as ollows. Pa ial Hadama d ma ices will be na u ally iden i ied as cliques o a sui able subg aph o I o’s Hadama d G aph [11]. This subg aph and i s p ope ies will be analyzed in Sec ion 2. F om his in o ma ion, in Sec ion 3 he p oblem o adding a new e ex o a gi en clique in G will be desc ibed as a Cons ain Sa is ac ion P oblem, o be sol ed by means o Minion [8]. Some examples will be p o ided. Las sec ion will be de o ed o conclusions and commen s abou u he wo k. A b ie ske ch o he wo k has been ecen ly exposed in [1], as a esul o a subs an ial p og ess on a p e ious wo k o some o he au ho s [2]. 2. The g aph G Hadama d G aphs we e in oduced by I o in [11]. O iginally hey e e ed o he g aph ∆(4 ) whose e ices a e he (1,−1)- ec o s o leng h 4 consis ing o an e en numbe o 1s. The adjacency ela ion consis s in o hogonali y. We call Hadama d g aph o he subg aph G o ∆(4 ) induced by he (1,−1)- ec o s simul aneously o hogonal o he h ee i s ows o a no malized Hadama d ma ix, ⎛ ⎜ ⎝ 1. . . 1 1 . . . 1 1 . . . 1 1 . . . 1 1. . . 1 1 . . . 1−1. . . −1−1. . . −1 1. . . 1−1. . . −1 1 . . . 1−1. . . −1 . . . . . . . . . . . . ⎞ ⎟ ⎠. These o hogonali y condi ions cha ac e ize s aigh o wa dly he o m o he e ices in G , as ollows. Lemma 1. The e ices o G consis o (1,−1)- ec o s o leng h 4 whe e he 2 nega i e en ies a e dis ibu ed so ha exac ly k, −k, −k and k nega i e en ies occu espec i ely among he anges [1, . . . , ],[ +1, . . . , 2 ],[2 +1, . . . , 3 ]and [3 +1, . . . , 4 ], o some 0≤k≤ . We may hen classi y he se o e ices in G a ending o he numbe ko nega i e en ies which appea in posi ions 1 o . In wha ollows, a k- e ex (o k- ec o ) in G e e s o a e ex wi h p ecisely knega i e en ies among posi ions 1 o . Ac ually, o he emainde o he pape , i can be assumed ha 0 ≤k≤ ⌊ 2⌋, since any ec o and i s nega ed − sha e a common se o adjacen e ices. I is eadily checked ha he size o G (bo h in e ices and edges) g ows exponen ially on . Lemma 2. In pa icula , he numbe o e ices in G is |G | = ∑ k=0( k)4 . He ea e , o b e i y, we will adop he addi i e no a ion o ep esen ing Hadama d ma ices, so ha he 1s u n o 0s and he −1s u n o 1s. This way, k- ec o s in G a e now desc ibed as (0,1)- ec o s o leng h 4 consis ing o p ecisely 2 ones (and hence 2 ze os), which a e dis ibu ed in he ollowing way: he e a e exac ly kones in posi ions 1 o , o he −k ones in posi ions +1 o 2 , ano he −kones in posi ions 2 +1 o 3 , he las kones being loca ed in posi ions 3 +1 o 4 . Each o hese k- ec o s may be s aigh o wa dly codi ied as an in ege , assuming ha he k- ec o is he bina y ep esen a ion o a decimal numbe . The e o e, cliques a e codi ied as lis s o in ege s, each o hem being he decimal ep esen a ion o a bina y numbe consis ing o 2 ones and leng h less o equal o 4 . Since cliques o size min G ansla e o pa ial Hadama d ma ices (m+3)×4 , we would like o sea ch o la ge cliques in G . Since he la ges clique in ∆(4 ) is a mos o size 4 , he la ges clique in G is a mos o size 4 −3. Cliques mee ing his uppe bound would co espond, in u n, o ull 4 ×4 Hadama d ma ices. Amaximum clique is a clique wi h he maximum ca dinali y (which is called he maximum clique numbe ). This no ion is di e en om ha o maximal clique, which e e s o a clique which is no a p ope subse o any o he clique. Thus, maximal cliques do no need o be maximum ones, hough he con e se is always ue. Conce ning cliques in G , his means ha a pa ial Hadama d ma ix does no need o be a subma ix o a ull Hadama d ma ix. Gi en a g aph, he maximum clique p oblem consis s in inding a maximum clique, and i is NP-comple e [3]. Fu he mo e, i isknown ha he eis nopolynomial- imealgo i hm o app oxima ing he maximum cliquewi hin a ac o o n1−ϵunless P =NP [9], whe e nis he numbe o he e ices o he g aph. Mo eo e , he e is no polynomial- ime algo i hm app oxima ing he clique numbe wi hin a ac o o n (log n)1−ϵunless NP =ZPP [12]. Fo una ely, he aim o he pape is no o desc ibe a gene al pu pose algo i hm o sol ing he maximum clique p oblem. The aspi a ion is o design an ad hoc algo i hm o cons uc ing su icien ly la ge cliques in G . To his end, we need o s udy he p ope ies o G in a mo e de ailed way. Fi s ly, we desc ibe a p ocedu e o de e mining a se δ o gene a o s o he adjacency lis ela ed o a ixed k- ec o , ha is, o hose s- ec o s wsha ing exac ly 2 en ies wi h . Lemma 3. A he i s and ou h ( esp., he second and hi d) qua e s he numbe i o coincidences in 1s ( esp., in 0s) uns in he ange i ∈ [0,min(k,s)]. Lemma 4. A each qua e , he numbe αio o al coincidences (bo h in 1s and 0s) sa is ies αi= −s−k+2i, and uns in he ange αi∈ [ −s−k, − |s−k|]. Co olla y 1. αi+1=αi+2. Le n=min(k,s). In he condi ions abo e, he se o o al coincidences is gi en by −s−k=α0<· · · < αn= −|s−k|. We may now desc ibe he se o s- ec o s adjacen o a gi en k- ec o . P oposi ion 1. The se o ec o s ha a e o hogonal o a gi en k- ec o co esponds o he ull se o dis ibu ions o ec o s sa is ying uples o o al coincidences (αi1, αi2, αi3, αi4)such ha αi1+αi2+αi3+αi4=2 . P oposi ion 2. The se o uples (αi1, αi2, αi3, αi4)which gi e ise o o hogonal s- ec o s a e cha ac e ized as he solu ions o he ollowing sys em o diophan ine equa ions {x0α0+. . . +xnαn=2 x0+. . . +xn=4 xi∈Z:0≤xi≤4 (2.1) He e, n =min(k,s)and xiindica es how many coincidences o he ype αimus occu among he ou qua e s. We now gi e a cons uc i e way o sol e he sys em abo e. P oposi ion 3. The e exis s a solu ion o he sys em (2.1) i and only i 4α0≤2 ≤4αn. Co olla y 2. Fixed and 0≤k,s≤ ⌊ 2⌋, he se sol o solu ions o he sys em (2.1) may be cons uc ed in he ollowing way: sol ← ∅ α0← −k−s αn← − |k−s| o i1 om max{α1,2 −3αn} o min{αn,⌊2 4⌋} wi h s ep 2 do o i2 om max{i1,2 −i1−2αn} o min{αn,⌊2 −i1 3⌋} wi h s ep 2 do o i3 om max{i2,2 −i1−i2−αn} o min{αn,⌊2 −i1−i2 2⌋} wi h s ep 2 do sol ←sol ∪ {{i1,i2,i3,2 −i1−i2−i3}} od od od Gi en a uple (αi1, αi2, αi3, αi4) solu ion o (2.1), cons uc he ou ma ices Nkwhose ows a e hose ec o s sa is ying αik o al coincidences wi h he co esponding qua e o . By cons uc ion, he jux aposi ion o any o he ows o hese ma ices gi es a ec o o hogonal o . P oposi ion 4. A se δ o gene a o s o he adjacency lis o may be s aigh o wa dly cons uc ed in e ms o ma ices o he ype abo e. P oposi ion 5. Fixed a k- ec o , he e exis s- ec o s o hogonal o i and only i s ∈ [⌈ 2⌉ − k,⌊ 2⌋]. Fu he mo e, we may s aigh o wa dly p ecise he numbe o s- ec o s o hogonal o a gi en k- ec o , o some ixed s∈ [⌈ 2⌉ − k,⌊ 2⌋]. Table 2.1 Rows in Hadama d ma ices a e s- ows, o s∈ {⌊ 2⌋ − 1,⌊ 2⌋}. 3 4 5 6 7 8 9 ow4 6 5 4 5 4 4 #{s= ⌊ 2⌋} 8 8 15 11 21 12 26 #{s= ⌊ 2⌋ − 1}0 4 1 9 3 16 6 Lemma 5. Fixed a alid dis ibu ion (i1,i2,i3,i4), he numbe o s- ec o s o hogonal o a gi en k- ec o is gi en by he exp ession: (k i1)( −k s−i1)( k i2)( −k s−i2)( k i3)( −k s−i3)( k i4)( −k s−i4) In pa icula , his migh sugges ha la ge cliques in G should mo e likely consis o k- ec o s, o la ge alues o k, close o ⌊ 2⌋. This seems o be he case in p ac ice, as he ollowing calcula ions sugges . Fo each 3 ≤ ≤9, we choose a andom a Hadama d ma ix o o de 4 om Sloane’s online lib a y [14], say had.12, had16.4,had20.hall.n,had24.pal,had28.pal2,had32.pal,had36.pal2. We now no malize hese ma ices, by means o he ollowing algo i hm. No ice ha since jus nega ion and pe mu a ion o columns a e used, he Hadama d cha ac e o he ma ix is p ese ed. Algo i hm 1. Hadama d no maliza ion •Nega e hose columns consis ing o a i s nega i e en y. •Now, loca e hose columns i, 1 ≤i≤2 , consis ing o a second nega i e en y. Simila ly, loca e hose columns j, 2 +1≤j≤4 , consis ing o a second posi i e en y. In e change hem. •P oceed as he s ep be o e, now by qua e s. As a esul , you will ob ain a no malized Hadama d ma ix. Fixed a ⌊ 2⌋- ec o among he ows o hese ma ices (as indica ed in Table 2.1), one may ha e a look on he ange in which he alues s un, o he s- ec o s de ining he emaining ows. Table 2.1 sugges s ha one should ocus on s- ec o s o s∈ {⌊ 2⌋ − 1,⌊ 2⌋}. We ake ad an age o his ac o speeding he algo i hm ex ending cliques in G , as desc ibed in he ollowing sec ion. 3. A CSP o ex ending cliques in G As we commen ed be o e, inding he maximum clique o a g aph is a NP-Ha d p oblem, and consequen ly exac algo i hms o his pu pose a e in easible e en in case o mode a ely la ge p oblem ins ances. The e o e mos o he e o s o gi e p ac ical solu ions o he maximum clique p oblem a e based on heu is ic app oaches. The au ho s hemsel es ied ou some o hem in a p elimina y wo k [2]. Un o una ely, no ma e he choice o he heu is ic is, all o hem equi e o explici ly cons uc he adjacency lis s o he e ices which a e conside ed along he compu a ion. And i is appa en om he p eceden sec ion ha such a ask is in ac able o he case o he g aph G , as soon as inc eases. We desc ibe he e a no el and comple ely di e en app oach. The idea is ansla ing he p oblem o ex ending a gi en clique in G , in o a Cons ain Sa is ac ion P oblem [6] (CSP in b ie , he ea e ). The cons ain sa is ac ion pa adigm pu sues sol ing a p oblem as he se o simul aneous solu ions o a se ies o cons ain s comple ely cha ac e izing he o me . As no iced in [6], ‘‘cons ain s iden i y he impossible and educe he ealm o possibili ies o e ec i ely ocus on he possible, allowing o a na u al decla a i e o mula ion o wha mus be sa is ied, wi hou exp essing how’’. Any model cha ac e izing a CSP consis s o a ini e se o a iables, hei ini e domains and he cons ain s o be sa is ied. As usual, i o en occu s ha he same p oblem may be modeled in di e en ways, each o hem in ol ing possibly di e en complexi ies. A ca e ul s udy o he pa icula ci cums ances o he p oblem should pe mi disc imina ing be ween hem. A solu ion o an ins ance o CSP is an assignmen o each a iable, such ha all cons ain s a e simul aneously sa is ied. Sol e s ypically ind all (o op ionally jus one) solu ions, i any does exis . In ou case, an explici o mula ion o he CSP elays on he knowledge o he p ope ies o G desc ibed in he p e ious sec ion. Mo e speci ically, gi en a clique Cin G , we look o an s- ec o (condi ion (C2) below) x=(x1,...,x4 ), o he wo mos p omising alues o s(condi ion (C1) below), which is simul aneously o hogonal o e e y e ex al eady in he clique (condi ion (C3) below). The e o e, a model o he CSP consis s in he ollowing cons ain s: (C1) ⌊ 2⌋ − 1≤s≤ ⌊ 2⌋. (C2) The numbe o −1s in he anges (x1, . . . , x ), (x +1, . . . , x2 ), (x2 +1, . . . , x3 ) and (x3 +1, . . . , x4 ) a e s, −s, −sand s, espec i ely. (C3) The numbe o coincidences o wi h each o he e ices al eady in Cis 2 . Once a model is ixed, he ollowing s ep consis s in ca ying i in o a sol e . A ansla ion o he cons ain s de ining he model in o he syn axis o he sol e is needed. In his pape we make use o Minion [8], one o he as es and mos scalable cons ain sol e s using he ‘‘model and un’’ me hodology. Ac ually, his sol e is based on he common echnique o al e na ing be ween spli ing and p opaga ion. The key poin he e is ha spli ing is minimized in p ac ice by e ec i e p opaga ion, since he o me inhe i s an exponen ial- ime solu ion me hod. Ne e heless, unlike mos o he cons ain sol e s, Minion does no b eak up cons ain s in o smalle pieces, nei he in oduce new a iables no simpli y o manipula e cons ain s. Typically, a dep h- i s ch onological back acking is pe o med by de aul . Bu ew u he in o ma ion is a ailable abou he in e nals o Minion’s pe o mance. In essence, i is a black box om he use poin o iew, delibe a ely p o iding ew op ions, bu gua an eeing aw speed in e u n. Minion expec s o be p o ided wi h he name o an inpu ile .min as an a gumen . This ile con ains a speci ica ion o he CSP o be sol ed (in e ms o a iables and cons ain s, o be decla ed in sepa a ed sec ions) as well as se ings ha he sea ch p ocess should use (a numbe o swi ches a e suppo ed o augmen de aul beha io ). The cons ain s sec ion consis s o any numbe o cons ain decla a ions on sepa a e lines, cons uc ed om he limi ed ange o elemen a y cons ain s which a e a ailable. Fou di e en a iable ypes may be decla ed. So ed by speed o pe o mance, boolean (BOOL) a iables ope a e as e , ollowed by disc e e (DISCRETE), delimi ed domains (BOUND), and a bi a y anges o in ege s (SPARSEBOUND). I no a iable o de ing is explici ly s a ed, hen one is gene a ed based on he o de he a iables a e decla ed. Ob iously, depending on he choice o he o de ing, he ime equi ed o inding a solu ion may a y in a signi ican way. The in e es ed eade is e e ed o [8,13] o mo e in o ma ion conce ning Minion. We now desc ibe he way in which he CSP o ex ending a gi en clique Cin G has been o malized in o Minion syn ax. Th ee a iables a e conside ed. The desi ed e ex x=(x1, . . . , x4 ) which migh po en ially ex end he clique Cis codi ied as a boolean ma ix xo 4× unknowns, he ow i ela ed o he ange o coo dina es x(i−1) +1, . . . , xi , o 1 ≤i≤4. Assuming he ad an ages commen ed in he p eceden sec ion, a disc e e a iable s DISCRETE s {⌊ 2⌋ − 1..⌊ 2⌋} p o ides s aigh o wa dly a simple way o ul ill he cons ain (C1). Condi ions (C2) a e codi ied in e ms o he cons ain occu ence( ec,elem,coun ), which ensu es ha he e a e coun occu ences o he alue elem in he ec o ec. This way, he se o cons ain s ocu ence([x[0,_]],1,s) ocu ence([x[1,_]],0,s) ocu ence([x[2,_]],0,s) ocu ence([x[3,_]],1,s) cha ac e izes ha he e ex xwhich we a e looking o de ines an s- ec o . Finally, ansla ing he ela ions (C3) equi es a mo e sophis ica ed elabo a ion, which combines he use o he cons ain elemen ( ec,i,e) and he cons ain ei y(cons, ). The o me speci ies ha , in any solu ion, ec[i] = e. The la e ensu es ha he boolean a iable is se o 1 i and only i he cons ain cons is sa is ied. Conside a boolean ma ix coin[|C|,4 ], whose i h- ow will keep ace o he en ywise coincidences o xand he i h e ex o C, so ha coin[i−1,j−1] = 1 i and only i x[⌊j−1 ⌋,(j−1) mod ] = C(i,j). As soon as he i h- ow o coin consis s o exac ly 2 ones, hen he e ex xwill be o hogonal o he e ex iin C. The e o e, he ela ions (C3) a e codi ied as |C|blocks o 4 +1 cons ain s o he ype: ei y(elemen (x[0,_],0,C(i,1)),coin[i−1,0]) . . . ei y(elemen (x[0,_], −1,C(i, )),coin[i−1, −1]) . . . ei y(elemen (x[3,_],4 −1,C(i,4 )),coin[i−1,4 −1]) ocu ence([coin[i−1,_]],1,2 ) Ob iously, i is a om being ope a i e ha all hese cons ain s a e in oduced by hand, s ep by s ep. Ne e heless, i is p e e able o make use o some p og amming language and design a small execu able ile which, p o ided Cas inpu da a, s aigh o wa dly gene a es he ull code o he Minion ile .min o be execu ed in u n. We ha e pe o med 10 sea ches o each in he ange 3 ≤ ≤11. In all cases, s a ing om a e ex andomly gene a ed, he p ocedu e has consis ed in i e a i ely ying o add a new e ex o he s uc u e al eady cons uc ed, un il he CSP ails o p o ide a new e ex. Table 3.2 esumes he a e age ime (A .T.) o each o hese calcula ions, as well as he smalles (Sm.) and la ges (Lg.) sizes o he cliques ound so a . No ice ha o small ull o almos ull Hadama d ma ices ha e been ound, and o la ge cliques ha e been ound a ound he bound 2 . 4. Conclusions and u he wo k In his pape a new way o gene a ing (possibly la ge) pa ial Hadama d ma ices m×4 has been desc ibed. Two a e he main no el ies o he app oach. On one hand, he p oblem has been ansla ed o he con ex o G aph Theo y, in ol ing he cons uc ion o cliques in ce ain subg aph G o I o’s Hadama d G aph ∆(4 ). Secondly, he p oblem i sel o ex ending a clique has been desc ibed as a Cons ain Sa is ac ion P oblem. Table 3.2 A summa y o he esul s, in ol ing abou 1500 uns o he p ocedu e. A .T. Sm. Lg. 3 1′′ 9 9 4 1′′ 12 13 5 1′′ 9 17 6 2′′ 9 21 7 8′′ 10 17 8 39′′ 12 21 9 7′50′′ 14 16 10 1h32′10′′ 15 17 11 8h38′9′′ 14 17 Al hough he size o he g aph G ce ainly makes he p oblem un ac able e en o no so la ge alues o , in compa ison o he wo k in [2], his app oach acili a es ex ending cliques o la ge alues o . Fu he mo e, o common alues o , i imp o es ei he he size o he ou pu clique, o he equi ed ime o execu ion, o e en bo h. As a ma e o ac , i is wo h no ing ha he app oach i sel migh be used o ex end hose cliques desc ibed in he in oduc ion, ounding he bound o 2 e ices, ob ained by he jux aposi ion o mos o he ows o wo ull Hadama d ma ices o app op ia ed sizes 1+ 2= . Fo ins ance, p og essing om he jux aposi ion o wo Hadama d ma ices o o de 16, i e a ion o he p ocedu e ends p o iding a ull Hadama d ma ix o o de 32. I akes ba ely 8 s o add a e ex in each s ep. Un o una ely, he sizes o he objec s which migh be o eal in e es ( hose ela ed o o de s o which no ull Hadama d ma ices a e known) a e ou o he scope o he ac ual capabili ies o which he communi y has access o, o he momen . Howe e , as a seconda y bene i , he p ocedu e i sel as de ined migh shed ligh on a new p ac ical way o a o d ha d p oblems on G aph Theo y, in e ms o Cons ain Sa is ac ion P oblems accu a ely designed ad hoc, a beyond he adi ional and didac ical use o he g aph colo ing p oblem o illus a ing he ypical pe o mance o a CSP. The doo is open o many o he in e es ing and po en ial applica ions. Re e ences [1] V. Ál a ez, J.A. A ma io, R.M. Falcón, M.D. F au, F. Gudiel, M.B. Güemes, A. Osuna, Gene a ing pa ial Hadama d ma ices as solu ions o a cons ain sa is ac ion p oblem cha ac e izing cliques. in: P oceedings o X Encuen o Andaluz de Ma emá ica Disc e a, pp. 17–20 ISBN: 978-84-697-4743-8, (La Línea, Cádiz, July 10–11 2017). [2] V. Ál a ez, J.A. A ma io, M.D. F au, F. 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