Probabilistic existence of rigid combinatorial structures

Probabilis tic existence of rigid combinatorial structures (extended abstract v ersion) Greg Kuperber g ∗ Shachar Lov ett † Ron Peled ‡ October 25, 2018 Abstract W e show the existence of rigid co mbinato rial objects which previously were n ot k nown to exist. Specifically , for a wide rang e of the und erlying parameters, we show the existence of non -trivial orthog- onal arrays, t -designs, and t -wise permu tations. In all cases, the sizes of the o bjects are optim al up to polyno mial overhead. The proof of existence is prob abilistic. W e show that a ran domly chosen such object has the requ ired prop erties with p ositiv e yet tiny prob ability . Th e main technical ingred ient is a special local central limit theorem for suitable lattice rando m walks with finitely many steps. 1 Introd uction W e introduce a ne w frame work for establis hing the ex istence of rigid combinatorial structur es, such as orthog onal arrays, t -d esigns and t -wise permutati ons. Let B be a finite set and let V be a v ector space of functi ons from B to the ration al numbers Q . W e study when there is a small subset T ⊂ B s atisfyin g 1 | T | ∑ t ∈ T f ( t ) = 1 | B | ∑ b ∈ B f ( b ) for all f in V . (1) In probab ilistic terminolo gy , equatio n (1) means that if t is a uniformly rando m element in T and b is a unifor mly random element in B then E [ f ( t )] = E [ f ( b )] for all f in V , (2) where E denotes exp ectati on. Of course, (1) holds trivial ly when T = B . Our goal is to find conditions on B and V that yield a small su bset T that satisfies (1), where in our situ ations, small will mean polyn omial in the d imension o f V . (In man y natura l proble ms one mig ht encoun ter a fu nction spa ce V ove r R or C instead. Ho wev er , since (1) is a rationa l equation, we can al ways reduce to the case of rational vect or spaces.) Our main theorem, Theorem 2.1, gi ve s sufficient condition s for the existenc e of a small sub set T sat- isfyin g (1 ). W e apply the theorem to estab lish results in three interesti ng cases of the general frame work : orthog onal arrays, t -desig ns, and t -wise permutat ions. These are detaile d in the next section s. Our methods solv e an open problem, w hether there exist non-tri vial t -wise permutations for ev ery t . They stre ngthen T eirlinck’ s theorem [T ei87], which was th e first theor em to sho w the exi stence of t -designs for ev ery t . And the y improve exist ence results for orthog onal arrays, when the size of the alphab et is divisi ble by many ∗ Univ ersit y of California, Davis. E-mail: greg@mat h.ucdavis. edu . Supported by NSF grant CCF-1013079. † Institute for Adva nced Study . E-mail : slovett@math.ias.ed u . Supported by NSF grant DMS-0835373 . ‡ T el A viv Un iv ersity , Israel. E-mail: peledron@post. tau.ac.il . Supported by an ISF grant and an IRG grant. 1 distin ct primes. Moreov er , in all three cases conside red, we show the existe nce of a structure whose size is optimal up to polyno mial ov erhead . Our approach to the problem is via probabilisti c argu ments. In essence, we prov e that a random subse t of B satisfies equation (1 ) with positi ve, albeit tiny , probab ility . Thus our method is one of the few known methods for sho wing existe nce of rare objects. This class inclu des such other method s as the Lov ´ asz local lemma [E L75] and Spencer ’ s “six de viati ons suf fice” m ethod [Spe85 ]. Ho wev er , our method does not rely on these previo us approaches . Instea d, our techni cal ingredient is a special versi on of the (multi- dimensio nal) local central limit theorem with only finitely many av ailable ste ps. S ince only finitely many steps are av ailab le, and since w e can only gain access to more steps by increasi ng the dimensio n of the random wal k, we canno t use an y “off the shelf ” local central limit theorem, not ev en one enhanc ed by a Berry-Essee n-type estimate of the rate of con ver gence . Instead, we prove the lo cal c entral li mit t heorem that we need direct ly using Fouri er analysis . Section 1.4 gi ves an o v ervie w of our approach. W e also mention that ef ficient randomiz ed algorithm versions of the Lov ´ asz local lemma [Mos09, MT10] and Spencer’ s method [Ban10] hav e recently been found. R elati ve to these new algorithms , the objects that the y produce are no longer rare. Our method is the only one that we kno w that sho ws the existence of rare combina torial structu res, which are still rare relati ve to any kno w n, efficie nt, randomized algorithm. 1.1 Orthogonal arrays A subset T ⊂ [ q ] n is an orthog onal arr ay of alphab et size q, length n and str engt h t if it yields all string s of lengt h t with equal freque ncy if restricted to any t coordinate s. In other words , for an y distin ct indices i 1 , . . . , i t ∈ [ n ] and an y (not necessar ily distinct) v alues v 1 , . . . , v t ∈ [ q ] , |{ x ∈ T : x i 1 = v 1 , . . . , x i t = v t }| = q − t | T | . Equi v alent ly , choosing x = ( x 1 , . . . , x n ) ∈ T uniformly , the distrib ution of x ∈ [ q ] n is t -wise ind epende nt. For an introdu ction to ortho gonal arrays see [HSS99]. Orthogon al arrays fit into our general framewo rk as follows. W e take B to be [ q ] n and V to be the space spann ed by all functio ns of the form f ( I , v ) ( x 1 , . . . , x n ) = ( 1 x i = v i for all i ∈ I 0 Otherwise , (3) with I ⊂ [ n ] a subset of size t and v ∈ [ q ] I . W ith this choice, a subset T ⊂ B satisfy ing (1) is precisely an orthog onal array of alphabe t size q , length n and strength t . It is well kno wn that if T ⊂ [ q ] n is t -wise independe nt then | T | ≥  cqn t  t / 2 for some univ ersal constant c > 0 (see, e.g., [Rao73]). Matching const ruction s of size | T | ≤ q c t  n t  c q t are kno wn, howe ver , as these rely on finite fi eld propert ies the constant c q genera lly tends to infinity with the number of prime fa ctors of q . Our technique prov ides the first upper bound on the size of orthogon al array s in which the constant in the exp onent is independ ent of q . Theor em 1.1 (Existe nce of orthogo nal arrays) . F or all inte ger s q ≥ 2 , n ≥ 1 and 1 ≤ t ≤ n ther e exist s an ortho gona l arr ay T of alphabet size q, length n and str ength t satisfying | T | ≤ ( qn ) c t for some univers al consta nt c > 0 . 1.2 Designs A (simple ) t - ( v , k , λ ) design is a family of distin ct subsets of [ v ] , where each set is of size k , such that each t elements belo ng to exa ctly λ sets. In oth er words, den oting by  v k  the f amily of all su bsets of [ v ] of size k , a 2 set T ⊂  v k  is a t -design if for any distin ct elements i 1 , . . . , i t ∈ [ v ] , |{ s ∈ T : i 1 , . . . , i t ∈ s }| =  k t   v t  | T | = λ . (4) For an in troduc tion to combinatori al designs see [CD07]. Our general frame work inc ludes t -designs as follo ws. W e take B to b e  v k  and V to be the space s panne d by all funct ions of the form f a ( b ) = ( 1 a ⊂ b 0 Otherwise , (5) with a ∈  v t  . W ith this choice, a subset T ⊂ B satisf ying (1) is precisely a simple t − ( v , k , λ ) design. Although t -designs hav e been in vestig ated for many years, the basic question of existenc e of a design for a giv en set of paramete rs t , v , k and λ remains m ostly unans wered unless t is quite small. The case t = 2 is kno wn as a block design and much more is kno wn about it than for lar ger t . Explicit constructi ons of t -designs for t ≥ 3 are kno wn for v arious spec ific constant settings of the parameter s (e.g. 5- ( 12 , 6 , 1 ) design ). The breakthroug h result of T eirlinck [T ei87] was the first to estab lish the existe nce of non-tri vial t -designs for t ≥ 7. In T eirlinck’ s construc tion, k = t + 1 and v satisfies congruen ces that grow v ery quickl y as a functi on of t . Other spora dic and infinite e xamples h a ve b een foun d since then (see [CD07] or [Mag09] and the references w ithin), howe ver , the set of parameters which they cov er is still ver y sparse. Moreo ve r , it follo ws from (4 ) that any t − ( v , k , λ ) design T has size | T | = λ  v t  /  k t  ≥ ( v / k ) t . Even when exist ence has been sho wn, the designs obtained are often inefficien t in the sense that their size is far from this lower bound . One of the main results of our work is to establish the existenc e of ef ficient t -desig ns for a w ide range of paramete rs. Theor em 1 .2 (Existen ce of t -design s) . F or all inte ger s v ≥ 1 , 1 ≤ t ≤ v and t ≤ k ≤ v t her e e xists a t - ( v , k , λ ) design whose size is at most v c t for some univer sal constant c > 0 . 1.3 Pe rmutations A family of permutations T ⊂ S n is called a t -wise permuta tion if its action on any t -tuple of elements is unifor m. In oth er words, for any disti nct elements i 1 , . . . , i t ∈ [ n ] and distinct elements j 1 , . . . , j t ∈ [ n ] , |{ π ∈ T : π ( i 1 ) = j 1 , . . . , π ( i t ) = j t }| = 1 n ( n − 1 ) · · · ( n − t + 1 ) | T | . (6) Our general framew ork includes t -wise permutatio ns as follo ws. W e take B = S n and V to be the space spann ed by all functio ns of the form f ( i , j ) ( b ) = ( 1 b ( i 1 ) = j 1 , . . . , b ( i t ) = j t 0 Otherwise , where i = ( i 1 , . . . , i t ) and j = ( j 1 , . . . , j t ) are t -tupl es of distinct element s in [ n ] . W ith this choice , a subset T ⊂ B satisfy ing (1) is precisely a t -wise permutatio n. Construc tions of famil ies o f t -wise permutations a re kno wn on ly for t = 1 , 2 , 3: the gr oup of cyclic shift s x 7→ x + a modulo n is a 1-wise permutat ion; the group of in verti ble affine transfo rmations x 7→ ax + b ov er a finite field F yields a 2-wise pe rmutation ; and the group of M ¨ obius tran sformatio ns x 7→ ( ax + b ) / ( cx + d ) with ad − bc = 1 ov er the projecti ve line F ∪ { ∞ } yields a 3-wise permutati on. For t ≥ 4 (an d n lar ge en ough) , ho we ve r , no t -wise permutation is known, other then the full symmetric group S n and the alternatin g group A n [KNR05, AL11]. In fact, it is known (c.f., e.g., [Cam95], Theorem 5.2) that for n ≥ 25 and t ≥ 4 there 3 are no other subgr oups of S n which form a t -wise permutation. (On other words, there are no other t - transit i ve su bgrou ps of S n for t ≥ 4 and n ≥ 25.) One of our main result s is to sho w e xistence of small t -wise permutat ions for all t . Theor em 1.3 (Existenc e of t -wise permutatio ns) . F or all inte ger s n ≥ 1 and 1 ≤ t ≤ n ther e exists a t -wise permutat ion T ⊂ S n satisfy ing | T | ≤ exp ( t c ) n c t for some unive rsa l constant c > 0 . It is clear f rom the de finition (6) abov e th at an y t -wise per mutation T must satisfy | T | ≥ n ( n − 1 ) · · · ( n − t + 1 ) = n Ω ( t ) . T hus, for fixed t , the t -wise per mutation s we exhibit are of optimal size up to polynomial ov erhea d. For t gro wing with n these t -wise permutatio ns may be la r ger , but stil l no lar ger than n t c for some uni versal constant c > 0. 1.4 Pr oof over view The idea of our app roach is as foll o ws. Let T be a random multiset of B of some fixed size N chosen by sampling B uniformly and independen tly N times (w ith replacement). Let ( φ a ) a ∈ A be a spanning set of inte ger -value d functions for V (where A is some finite inde x set). O bserv e that T satisfies (1) if and only if ∑ t ∈ T φ a ( t ) = N | B | ∑ b ∈ B φ a ( b ) = E " ∑ t ∈ T φ a ( t ) # for all a in A . (7) Thus defining an inte ger -v alued random v ariabl e X a : = ∑ t ∈ T φ a ( t ) and X : = ( X a ) a ∈ A ∈ Z A we see that existen ce of a subset of size N sati sfying (1) will follo w if we can sho w that P [ X = E [ X ]] > 0. T o this end we examine more closely the distrib ution o f X . Let t 1 , . . . , t N be the random elements chosen in forming T . The spannin g set ( φ a ) a ∈ A defines a mapping φ : B → Z A by the tri vial φ ( b ) a : = φ a ( b ) . Observ e that our choice of random model implies that the vectors ( φ ( t i )) i ∈ [ N ] are indepen dent an d ident ically distrib uted. Hence, X = ∑ i φ ( t i ) (8) may be vie w ed as the end po sition of an N -step random wa lk in th e lat tice Z | A | . Thus we may hope t hat if N is sufficie ntly lar ge, then X has an approximat ely (multi -dimensi onal) Gaussi an distrib ution by the central limit theorem. If the relev ant loca l central limit theorem holds as well, the n the pro babili ty P [ X = x ] also satisfies a Gaussian approximation . In particu lar , since a (non-d egen erate) Gaussia n always has positi ve densit y at its ex pectati on, we could conclude that P [ X = E [ X ]] > 0 as desired. The abov e descrip tion is the essence of our approach. The m ain obstacl e is, of course, point ed out in the last step. W e must control the rat e of con ver genc e of the loc al central limit the orem well enough that the con ver genc e error does not o utweigh the pr obabil ity densit y of the Gaus sian distrib ution at E [ X ] . Recall that the order of magnitude of such a density is typical ly c −| A | for some constant c > 1, and recall that | A | is at lea st the dimension of V , which is the main parameter of our problem. So w e indeed ha ve very small probabilit ies. For this rea son, and becau se we want con ver gence w hen N is only pol ynomial in the dimensio n of V , we w ere unable to use any standard local central limit theorem. Instea d, w e dev elop an ad hoc ver sion using direct Fourier anal ysis. In our proof of the m ain theorem, we modify the abo ve descripti on in one respect. It is technica lly more con venient to work with a slig htly diff erent probabi lity m odel. Instead of ch oosin g T as abov e, we set 4 p : = N / | B | and define T by taking each element of B into T indep enden tly w ith probabi lity p . This ha s the benefit of guaranteei ng that T is a pro per set instead of a multiset. Ho we ver , it has also the disadv antage that it does not guaran tee that | T | = N . T o remedy this, we assume that the space V contai ns the constant functi on h ( b ) = 1; o r i f n ot, we can add it to V at the minor cost o f inc reasing the dimensi on of V by 1. W ith this assumpt ion, we note that E  ∑ t ∈ T h ( t )  = E [ | T | ] = N . Thus (7), or equiv alently X = E [ X ] , also implies that | T | = N as required. A nother disadv antage is that in this ne w probability model, the vecto r X is no longer a sum of identical ly distr ib uted vari ables. Ho we ve r , since the summands in (8) are still independe nt, we can con tinue to use Fourier ana lysis methods in our proof. W e cannot expect there to alway s be a small subset T that satisfies (1). For instanc e, Alon and V u [A V97] found a regular hyp er graph with n vertices and ≈ n n / 2 edges, with no re gular sub -hyper graph. Here, the deg ree of a verte x is the number of hyperedges inciden t to it and a regular hyper graph is one in w hich the deg rees of all vertices are equal. W e m ay describ e their example in our language by letting B be the set of edges of this hyper grap h, A be its vertex set, and define φ : B → { 0 , 1 } A by lett ing φ ( b ) be the indicator functi on of the set of v ertice s incident to b . The result of [A V97] implies that while the vector ∑ b ∈ B φ ( b ) is constan t, this property is not shared by ∑ t ∈ T φ ( t ) for any non-empty , proper subset T ⊂ B . Thu s, we need to impose certain condi tions on B and V , or equiv alently on the map φ . W e start by requirin g certain di visib ility , boundedn ess and symmetry assumptions . Divisi bility: N is such that N | B | ∑ b ∈ B φ ( b ) is an integer ve ctor . This property is clearly necessary for (7) to hold and is typical ly a mild restric tion on N . Boundedness: The entrie s of φ must be small. M ore precisely , max a ∈ A , b ∈ B | φ ( b ) a | is bounde d by a poly- nomial in dim V , since our method requires N to b e at least some polynomia l in this maximum. Symmetry: A symmetry of φ is a pair consisti ng of a permutation π ∈ S B and a linear transformati on τ ∈ GL ( V ) which satisfies φ ( π ( b )) = τ ( φ ( b )) for all b ∈ B . The set of symmetries ( π , τ ) of φ is a subgro up of S B × GL ( V ) . W e re quire that the projecti on to B of the grou p of symmetries is transiti ve. In other words , that for any b 1 , b 2 ∈ B there exi sts a symmetry ( π , τ ) of φ satisfy ing π ( b 1 ) = b 2 . It is not hard to verify that the third condi tion is intrin sic to the structure of V and does not depen d on the specific choice of spanning set ( φ a ) . In our applica tions it follo ws easily from the overal l symmetry of the setup. Ho wev er , we also ha ve a fourth assumpt ion which is more techn ical than the others . First, we requir e that ( φ a ) a ∈ A forms a b asis of V . This implies that for an y a ∈ A , we may ex press e a , the un it vec tor with 1 at its a ’ th coo rdinate , as a linea r combina tion of the fo rm ∑ b ∈ B c b φ ( b ) . W e c all an y such l inear combinat ion an isolat ing co mbinatio n fo r a . W e assume that for each a ∈ A , there are many isolating combinations suppo rted on disjoin t subsets of B . Moreo ver , we require the coefficien ts of these combinat ions to hav e small norm and to be rational with a small common denomina tor . This is the most dif ficult assumpti on to verify in our applic ations. Sectio n 2 giv es more detail s about all of these assumptio ns. Our main theorem sho ws that these four conditio ns yield the existe nce of a small solution of (1). Theor em (Main theore m - in formal sta tement) . L et B b e a finite se t and let V be a vector s pace of functi ons fr om B to Q which contain s the con stant function s. If ther e exist s a ba sis ( φ a ) a ∈ A of V , co nsistin g of inte ger - valued function s, w hich satisfies the bounde dness , symmetry and isolat ion condit ions abo ve. Then ther e is a small subset T ⊂ B such tha t 1 | T | ∑ t ∈ T f ( t ) = 1 | T | ∑ b ∈ B f ( b ) 5 for all f in V . W e note that the size N = | T | of the subse t obtained m ust satisfy the div isibil ity con dition above . The exi stence theore ms for orthogonal arrays, t -designs and t -wise permutations follo w by showin g that for the choice of B and V detailed in S ection s 1.1 through 1.3 there exis ts a choice of basis { φ a } and small N for which all four conditi ons above hold . 1.5 Related work In the probabilisti c formulatio n (2) of our problem w e seek a small subset T ⊂ B such that the uniform distrib ution over T simulates the uniform distrib ution over B with regards to certain tes ts. There are two ways to rela x the problem to m ake i ts solution easier , and raise ne w questio ns reg arding expl icit solutions . One relaxatio n is to allo w a set T w ith a non-unifo rm distrib ution µ . For many practical applications of t -desig ns and t -wise permutations in statistics and computer science, but not quite ev ery application, this relaxa tion is as good as the uniform question. The existenc e of a solution with small suppor t is guaranteed by Carath ´ eodory’ s theorem, using the fact that the const raints on µ are all linea r equalities and inequalities . Moreo ver , such a solution can be found ef ficiently , as was shown by Karp and Papadimitriou [KP82] and in more general settings by Ko ller and Megidd o [KM94]. Alon and Lov ett [AL 11] giv e a strongly explicit analog of this in the case of t -w ise permut ations and m ore gene rally in the case of group actions. A dif ferent relaxation is to require the uniform distrib ution on T to only approx imately satisfy equation (2). Then it is tri vial that a suffici ently lar ge rand om subse t T ⊂ B satisfies the requiremen t with high probab ility , and the question is to find an expli cit solution. Fo r ins tance, we can relax the problem of t -wise permutat ions to almost t -wise permutatio ns. For this varian t an optimal solutio n (up to polynomial factors) was achie ved by Kaplan , Naor and Reingold [KNR05], who gav e a constructio n of such an almost t -wise permutat ion of size n O ( t ) . Alternati vely , one can start with the constant siz e expandin g set of S n gi ve n by Kassabo v [Kas07] and take a random walk on it of leng th O ( t log n ) . 1.6 Paper organization W e gi ve a precise descr iption of the general frame work and our main theorem in Section 2 . W e apply it to sho w the exis tence of orthogona l arrays a nd t -designs in S ection 3. T he case of t -wise pe rmutation s requires a detour to the representati on theory of the symmetric group, and we defer it to the full version of this paper . The proof of our main theorem is gi ven in Section 4. W e summarize and giv e some open proble ms in Section 5. 2 Main Theorem Let B be a finite set and let V be a ve ctor sp ace of function s from B to Q . W e ask for conditio ns for the exi stence of a small set T ⊂ B for w hich (1) hol ds. O ur theor em uses the follo wing notation . For a basis ( φ a ) a ∈ A (where A is some finite ind ex se t) of V w e d efine φ : B → Z A by φ ( b ) a = φ a ( b ) . This definitio n is e xtend ed linearly to φ : Z B → Z A by sett ing φ ( γ ) = ∑ b ∈ B γ b φ ( b ) . In the same manner , a set T ⊂ B is ident ified w ith its indicator vector so that φ ( T ) = ∑ t ∈ T φ ( t ) . Finally , we recall from Section 1.4 that a symmetry of φ is a pair π ∈ S B and τ ∈ GL ( V ) such that φ ( π ( b )) = τ ( φ ( b )) for all b in B . W e now state formally our main theorem. Theor em 2.1 (Main Theorem) . Let B be a finite set and V be a vector spa ce of functio ns fr om B to Q which contai ns the constant functions. Suppose that ther e e xist inte gers m , c 0 ≥ 1 , r eal numbers c 1 , c 2 , c 3 > 0 and a basis ( φ a ) a ∈ A of V consisting of inte ger -valu ed funct ions such that: 6 Divisi bility: c 0 | B | φ ( B ) is an inte ger vector . Boundedness: k φ ( b ) k 2 ≤ c 1 for all b ∈ B. Symmetry: F or ea ch b 1 , b 2 ∈ B ther e exists a symmetry ( π , τ ) of φ suc h that π ( b 1 ) = b 2 . Isolat ion: F o r any a ∈ A ther e exis t vector s γ 1 , . . . , γ r ∈ Z B for r ≥ | B | / c 2 suc h that • φ ( γ i ) = m · e a for all i ∈ [ r ] . • The vector s γ 1 , . . . , γ r have disjoint suppo rts, wher e the support of a vector γ ∈ Z B is the set of coor dinates on which i t is nonzer o. • k γ i k 2 ≤ c 3 for all i ∈ [ r ] . Then ther e exists a subset T ⊂ B with | T | ≤ poly ( | A | , m , c 0 , c 1 , c 2 , c 3 ) suc h that 1 | T | ∑ t ∈ T f ( t ) = 1 | B | ∑ b ∈ B f ( b ) for all f in V . W e prove Theorem 2.1 in Section 4. A careful examinatio n of the proof sho ws that we can choose | T | = N for any N ≥ 1 which satisfies the follo wing constra ints: • c 0 m di vide s N ; • N ≥ Ω ( 1 ) · max ( m 3 , | A | 2 m 2 log 2 ( | A | mc 0 c 1 c 2 c 3 ) , | A | 6 c 6 1 c 3 2 c 6 3 log 3 ( | A | mc 0 c 1 c 2 c 3 )) ; • N ≤ O ( p | B | ) . Of course , if the parameters are so large so that the second and third conditions contradict each other , then our theore m remains tri viall y true by taking T = B . 3 A pplications In th is sec tion we apply our main theo rem, Theorem 2.1, to pr ov e th e e xisten ce results fo r ort hogon al arrays and t -desi gns, Theorems 1.1 and 1.2. The exi stence res ult for t -wise permutations, Theorem 1.3, is more complica ted because it requires a discussion of the representati on theo ry of the symmetric group. W e defer it to the full versi on of this paper . 3.1 Orthogonal arrays W e use the choice of B and V described in Section 1.1 and recall the definition (3) of the function s f ( I , v ) of that secti on. W e note that for ev ery subse t I we hav e ∑ v ∈ [ q ] I f ( I , v ) ≡ 1. Thus V contains the constant fu nction s as Theorem 2.1 requires. W e start by choosing a con veni ent basis for V of integ er -v alued functions. Recall that the alphabet is [ q ] = { 1 , . . . , q } and let [ q − 1 ] = { 1 , . . . , q − 1 } be all symbols other than q . Extend the definitio n (3) of f ( I , v ) to apply to all subsets I with | I | ≤ t and v ∈ [ q ] I . Her e, w e mean that f ( / 0 , / 0 ) is the consta nt functio n 1. Finally , let A : = { ( I , v ) : | I | ≤ t , v ∈ [ q − 1 ] | I | } and for a = ( I , v ) ∈ A set φ a : = f ( I , v ) . Claim 3.1. The span of the functio ns { φ a } a ∈ A is V . 7 Pr oof. C learly φ a ∈ V for all a ∈ A . T o see that { φ a } a ∈ A spans V , we will show that any f ( I , v ) with | I | ≤ t and v ∈ [ q ] I is spann ed by { φ a } a ∈ A . W e do th is by ind uction on th e number of e lements in v which are equa l to q . First, if v ∈ [ q − 1 ] I then f ( I , v ) = φ ( I , v ) . Otherwise, let I = { i 1 , . . . , i r } with r ≤ t , v ∈ [ q ] I and assume WLOG that v i 1 = q . Then f ( I , v ) = f ( { i 2 ,..., i r } , ( v i 2 ,..., v i r )) − q − 1 ∑ m = 1 f ( I , ( m , v i 2 ,..., v i r )) and by inducti on, the right hand side belon gs to the linear span of { φ a } a ∈ A . Recall tha t φ : B → Z A is de fined as φ ( b ) a = φ a ( b ) . W e no w choos e intege rs m , c 0 ≥ 1 and rea l numbers c 1 , c 2 , c 3 > 0 such that the conditi ons of divisi bility , bounded ness, symmetry and isolat ion requ ired b y Theo- rem 2.1 are satisfied . F irst, le t a = ( I , v ) ∈ A . Note th at 1 | B | φ ( B ) a = q −| I | . Thus we set c 0 = q t so th at c 0 | B | φ ( B ) is an inte ger vector . Second , we clearly ha ve for any b ∈ B that k φ ( b ) k 2 2 = ∑ t i = 0  n i  ≤ ( n + 1 ) t . Hence we set c 1 = ( n + 1 ) t / 2 . Third, to witne ss the symmetry conditi on, fix x ∈ [ q ] n and consi der the permutation π ∈ S B gi ve n by π ( b ) = b + x ( mod q ) . W e need to sho w that there exists a linear map τ acting on V such that φ ( π ( b )) = τ ( φ ( b )) for all b ∈ B . This holds since for a = ( I , v ) ∈ A we ha v e φ ( π ( b )) a = f I , v ( b + x (mod q )) = f I , v − x (mod q ) ( b ) and f I , v − x (mod q ) ∈ V is in th e linear span of { φ a } a ∈ A by Claim 3.1. The fou rth cond ition we ne ed to v erify is the exi stence of many disjoin t isolation vec tors for each a ∈ A . Note that this conditi on al so implies that { φ a } a ∈ A is a basis fo r V . T his is establ ished in the following lemma. Lemma 3.2. Let a ∈ A. Ther e ex ist disjoint vector s γ 1 , . . . , γ r ∈ Z B with r ≥ | B | / ( q t n 2 t ) and k γ i k 2 ≤ 2 3 t / 2 n t suc h that φ ( γ i ) = e a . W e prov e Lemma 3.2 in two steps. First we fix some notations. Let K ⊂ [ n ] be of size | K | ≤ t , and let K c = [ n ] \ K . For x ∈ [ q ] n let x | K ∈ [ q ] K be the restri ction of x to the coordin ates of K . Abusin g notation, we also thi nk of x | K ∈ [ q ] n by setti ng coordin ates outside K to zero. Note that in this nota tion, f I , v ( x ) = 1 { x | I = v } . W e define the vecto r δ x , K ∈ Z B as δ x , K : = ∑ J ⊆ K ( − 1 ) | K |−| J | e x | J ∪ K c , where we recall that for b ∈ B , e b ∈ { 0 , 1 } B is the corre spondi ng unit vector . Note that if K = / 0 the n δ x , / 0 = e x . Claim 3.3. Let a = ( I , v ) ∈ A. T hen φ ( δ x , K ) a = ( 0 if K 6⊆ I 0 if a | K 6 = x | K 1 if a | K = x | K Pr oof. W e compute the v alue of φ ( δ x , K ) in coordin ate a = ( I , v ) ∈ A . W e ha ve φ ( δ x , K ) a = ∑ J ⊆ K ( − 1 ) | K |−| J | 1 { ( x | J ∪ K c ) | I = v } . Suppose first that K 6⊆ I . Then there exi sts j ∈ K \ I . Flipping the j -th element in J doesn’ t change the exp ression 1 { ( x | J ∪ K c ) | I = v } and hence the alter nating sign sum cancels. W e th us assume from no w on that K ⊆ I . W e thus ha ve 1 { ( x | J ∪ K c ) | I = v } = 1 { x | J = v | K and x | K c ∩ I = v | K c } . This ex pressio n ev aluates to 1 only if J = K and x | I = v . 8 W e ne xt prov e Lemma 3.2, sho w ing that we can b uild man y di sjoint isola tion v ector s for an y a ∈ A . The proof uses the vect ors δ x , K we just analyz ed. Pr oof of Lemma 3.2. F ix a = ( I , v ) . Let x ∈ [ q ] n be such that x | I = v . W e will construct a vec tor γ x , I such that φ ( γ x , I ) = e a . W e will do so by backward induc tion on | I | ≤ t . If | I | = t we take γ x , I : = δ x , I , and if | I | < t we construct recursi ve ly γ x , I : = δ x , I − ∑ K ) I , | K |≤ t , x K ∈ [ q − 1 ] K γ x , K . It is easy to ver ify using Claim 3.3 that indeed φ ( γ x , I ) = e a as claimed. W e further claim that k γ x , I k 2 ≤ 2 t / 2 ( 2 n ) t −| I | . This clearly holds if | I | = t . If | I | < t we bound by induction k γ x , I k 2 ≤ k δ x , I k 2 + t ∑ k = | I | + 1 ∑ K ⊃ I , | K | = k k γ x , K k 2 ≤ 2 t / 2 1 + t ∑ k = | I | + 1  n − | I | k − | I |  ( 2 n ) t − k ! ≤ 2 t / 2 1 + t ∑ k = | I | + 1 n k −| T | ( 2 n ) t − k ! ≤ 2 t / 2 n t −| I | 1 + t ∑ k = | I | + 1 ( 2 ) t − k ! = 2 t / 2 ( 2 n ) t −| I | . T o conclu de, w e need to sho w that by choosin g dif feren t v alue s for x such that x | I = v we can achie ve many disjoint vecto rs which isolate a . T he key observ ation is that γ x , I is supporte d on elements b ∈ B whose hamming distan ce from x is at most t . Thus, if we choose x 1 , . . . , x r ∈ [ q ] n such that ( x i ) | I = v and such that the hamming distance between each pair x i , x j is at least 2 t + 1, we get that γ x 1 , I , . . . , γ x r , I ha ve disjoi nt supports. W e can achie ve r ≥ q n − t / n 2 t by a simple greedy proce ss: choose x 1 , . . . , x r iterati vely; after choosi ng x i delete a ll ele ments in [ q ] n whose h amming d istanc e from x i is at most 2 t . Since th e numbe r of these elements is boun ded by ∑ 2 t i = 1  n i  ≤ n 2 t the claim follo w s. W e now ha ve all the condition s to apply Theorem 2.1. W e hav e | A | = ∑ t i = 0  n i  ( q − 1 ) i ≤ ( q ( n + 1 )) t , c 0 = q t , c 1 = ( n + 1 ) t / 2 , c 2 = q t n 2 t , c 3 = 2 3 t / 2 n t and m = 1. Hence we obtain that there exists an orthogo nal array T ⊂ [ q ] n of stren gth t and size | T | ≤ ( qn ) c for some uni ver sal constant c > 0. 3.2 Designs In this section , we prov e Theorem 1.2. It suf fices to prov e the theorem for k > 2 t , since if k ≤ 2 t then the complete design (the design containing all subsets of size k ) establish es the theorem. W e use the choice of B and V described in Section 1.2 and recall the definitio n (5) of the fun ctions f a of that section. W e set A =  v t  and note that ∑ a ∈ A f a ≡  k t  and thus V contain s the constan t function s as Theorem 2.1 requires. As a con venien t basis for V of integer -valu ed func tions, we take { φ a } a ∈ A with φ a = f a . By definitio n, { φ a } a ∈ A spans V and the fact that { φ a } a ∈ A is a basis for V will be implied by sho wing the isola tion condition of Theorem 2.1. W e choose inte gers m , c 0 ≥ 1 and real numbers c 1 , c 2 , c 3 > 0 to satisfy the conditi ons of di visib ility , bound edness , symmetry and i solatio n in Theorem 2.1. First, 1 | B | φ ( B ) =  k t  /  v t  · ( 1 , . . . , 1 ) and hence we set 9 c 0 =  v t  so th at c 0 | B | φ ( B ) is an int ege r vector . S econd, k φ ( b ) k 2 2 ≤ | A | ≤ v t . Hence we set c 1 = v t / 2 . Third, the symmetry co ndition also follo ws simply: let σ ∈ S v be a permutat ion on [ v ] . It acts nat urally on B an d A (b y permutin g subsets of [ v ] ) and giv es two permutatio ns π ∈ S B and ˜ π ∈ S A that satisfy φ ( π ( b )) a = φ ( b ) ˜ π − 1 ( a ) . The linear transfo rmation τ ∈ GL ( V ) then correspo nds to the permutation ˜ π − 1 . Finally , we need to sho w that for each a ∈ A there ex ist many disjoint vecto rs which isolate it. This is accompli shed in the follo wing lemma. Lemma 3.4. Assume k > 2 t . F or any a ∈ A ther e exist vectors γ 1 , . . . , γ r ∈ Z B with r ≥ | B | / ( vk ) 2 t suc h that φ ( γ i ) = k ! ( k − t ) ! · e a . Mor eo ver , γ 1 , . . . , γ r have disjo int supports and k γ i k 2 ≤ ( 2 k ) 3 t / 2 for i ∈ [ r ] . W e will need the follo wing technica l cla im for the proof of Lemma 3.4. In the follo w ing we consider binomial coef fi cients  n m  = 0 whene ver n < m . Claim 3.5. Let a > b ≥ 0 and c ≥ 0 . Then a ∑ i = 0 ( − 1 ) i  a i  c + i b  = 0 . Pr oof. L et f ( a , b , c ) = ∑ a i = 0 ( − 1 ) i  a i  c + i b  . If b , c > 0 we hav e  c + i b  =  c − 1 + i b  +  c − 1 + i b − 1  and hen ce f ( a , b , c ) = f ( a , b , c − 1 ) + f ( a , b − 1 , c − 1 ) . So, it is enoug h to veri fy the claim whene ver b = 0 or c = 0. If b = 0 then f ( a , 0 , c ) = ∑ a i = 0 ( − 1 ) i  a i  = 0 since a ≥ 1. If c = 0 then f ( a , b , 0 ) = ∑ a i = b ( − 1 ) i  a i  i b  =  a b  ∑ a i = b ( − 1 ) i  a − b i − b  = 0. Pr oof of Lemma 3.4. L et a ∈ A =  v t  be a co ordina te we wish to isola te. L et x ∈  v k  be a set di sjoint from a and let 0 ≤ j ≤ t . Define δ x , a , j ∈ Z B to be the indi cator vector for all subs ets b ∈ B =  v k  such that b ⊂ a ∪ x and | a ∩ b | = j , that is δ x , a , j : = ∑ b ⊂ a ∪ x , | b | = k , | a ∩ b | = j e b . W e define vecto rs γ a , x ∈ Z B as γ x , a : = t ∑ j = 0 ( − 1 ) t − j j ! ( k − j − 1 ) ! ( k − t − 1 ) ! δ x , a , j . W e will shortly sho w that φ ( γ x , a ) = k ! ( k − t ) ! e a . First we boun d the norm of γ x , a and sho w the exist ence of many disjoint v ectors . It is easy to check that k γ x , a k 2 ≤ ( 2 k ) 3 t / 2 . Also, the vector γ x , a is supported on coordinat es y ∈ B such that | y ∩ x | ≥ k − t . Thus, if we choose x 1 , . . . , x r ∈ B such that | x i ∩ x j | ≤ k − 2 t − 1 we get that the vec tors γ x 1 , a , . . . , γ x r , a ha ve disjoint suppo rt. W e can choose r ≥ | B | / ( vk ) 2 t by a simple greedy ar gument: choose x 1 , . . . , x r iterati vely , where in each step afte r choos ing x i we remove all subsets y ∈ B whose interse ction with x i is at lea st k − 2 t . The number of subset s eliminated in each step is at most ( vk ) 2 t hence we will get r ≥ | B | / ( vk ) 2 t . T o conclude the proof, w e need to compute φ ( γ x , a ) . L et a ′ ∈ A . Clearly if a ′ 6⊆ a ∪ x then φ ( γ x , a ) a ′ = 0. W e thus assume that a ′ ⊂ a ∪ x . Let ℓ = | a ∩ a ′ | w here 0 ≤ ℓ ≤ t . W e hav e that φ ( δ x , a , j ) a ′ = 0 if j < ℓ , and that φ ( δ x , a , j ) a ′ = |{ b ∈ B : a ′ ⊂ b ⊂ a ∪ x , | a ∩ b | = j }| =  t − ℓ t − j  k − t + ℓ j  . 10 Hence we ha ve that φ ( γ x , a ) a ′ = ( k − 1 ) ! ( k − t − 1 ) ! t ∑ j = ℓ ( − 1 ) t − j  t − ℓ t − j  k − t + ℓ j   k − 1 j  (9) If a ′ = a the n φ ( γ x , a ) a = k ! / ( k − t ) ! as claimed. T o conc lude we need to prov e that if a ′ 6 = a then φ ( γ x , a ) a ′ = 0. W e ha ve ℓ = | a ∩ a ′ | < t and let s = t − ℓ > 0. T hus φ ( γ x , a ) a ′ = ( − 1 ) t ( k − 1 ) ! ( k − t − 1 ) ! t ∑ j = ℓ ( − 1 ) j  t − ℓ t − j  k − t + ℓ j   k − 1 j  = ( − 1 ) s ( k − 1 ) ! ( k − t − 1 ) ! s ∑ j = 0 ( − 1 ) j  s j  k − s j + ℓ   k − 1 j + ℓ  = ( − 1 ) s ( k − 1 ) ! ( k − t − 1 ) !  k − 1 s − 1  s ∑ j = 0 ( − 1 ) j  s j  k − ℓ − 1 − j s − 1  = ( − 1 ) s ( k − 1 ) ! ( k − t − 1 ) !  k − 1 s − 1  s ∑ j = 0 ( − 1 ) j  s j  k − ℓ − 1 − s + j s − 1  . W e no w apply Claim 3.5 with a = s , b = s − 1 , c = k − ℓ − 1 − s a nd conclud e that φ ( γ x , a ) a ′ = 0. W e are no w ready to apply T heorem 2.1. W e ha ve | A | =  v t  , c 0 =  v t  , c 1 = v t / 2 , c 2 = ( vk ) 2 t , c 3 = ( 2 k ) 3 t / 2 and m = k ! / ( k − t ) !. Thus th e the orem implies t he e xiste nce of a t − ( v , k , λ ) desig n T ⊂ B with | T | ≤ v c t for some uni ve rsal constant c > 0. 4 Pr oof of Main Theorem W e prov e Theorem 2.1 in this sec tion. W e recall the settings: B is a finite set and V is a vecto r spac e of functi ons from B to Q . W e assu me the space V is spanned by integer value d func tions { φ a : B → Z } a ∈ A , where A is a finite inde x set. W e also assume that the constant functions belong to V . The proof strategy is conceptual ly simple: choose T randomly and sho w that this choice is succ essful with positi ve probabil ity . Let N be the targe t size of T , to be chosen later . Let each b ∈ B be chosen to be in T indepen dently w ith probabilit y p : = N / | B | . Identifyin g T w ith its indicato r vector in { 0 , 1 } B , we hav e that T b ∈ { 0 , 1 } with P [ T b = 1 ] = p . D efine X = φ ( T ) ∈ Z A and note that E [ X ] = p · φ ( B ) . In order to prov e Theorem 2.1 we need to sho w that P [ X = E [ X ]] > 0 . (10) W e make two notes: first, since we assume that consta nt functio ns belong to V we ha ve that if X = E [ X ] then in particular | X | = p | B | = N . Second , in order for (10) to hold we m ust hav e that E [ X ] is an inte ger vec tor . Thus, we must choose N to be di visible by c 0 . The difficul ty with establishi ng (10) comes from the fact that we require A diff erent e ven ts to occur simultan eously : for all a ∈ A we require that X a = E [ X a ] . T o better exp lain the challenge, consider momen- tarily fo r simpl icity the case whe re φ ( b ) ∈ { 0 , 1 } A for a ll b ∈ B and that f or ea ch a ∈ A , P b ∈ B [ φ ( b ) a = 1 ] = q (that is, all columns of φ ha ve qB ones) . Then each indi vidual X a is binomially distrib uted , X a ∼ Bin ( | B | , pq ) , and it is not hard to see that P [ X a = E [ X a ]] ≈ 1 √ qN . Ho wev er , we need the ev ents X a = E [ X a ] to occur simultaneo usly for all a ∈ A . T he problem arises becaus e these ev ents are dependen t, and gener al tec hnique s for handling su ch depen dencie s (for ex ample, the Lov ´ asz 11 local lemma) only work when each ev ent depends onl y on a few other ev ents (which is not the case here) and where eac h eve nt holds with suf fi ciently high prob ability (which is also not the case here). What we sho w is that, unde r t he condit ions of Theorem 2.1, if we choos e N large enoug h (b ut on ly polyno mially lar ge in | A | , m , c 0 , c 1 , c 2 , c 3 ) then all the e vent s X a = E [ X a ] become essentia lly independent , and we sho w that P [ X = E [ X ]] ≈ ∏ a ∈ A P [ X a = E [ X a ]] ≈ ( 1 √ qN ) | A | . The actual expre ssion we get is some w hat more complica ted as it also in volv es pair wise correlatio ns be tween the dif feren t eve nts X a , bu t conceptu ally it is of a similar flav or . Our main techniq ue to study the distrib ution of the random vari able X ∈ Z A is Fourie r analysis . W e recall some basic fact s about Fourier analys is on Z A . Fac t 4.1 (Fourier analysis on Z A ) . Let X ∈ Z A be a ran dom variable . The F ourier coef ficien ts of X live in the A -dimensio nal torus. Let T = [ − 1 / 2 , 1 / 2 ) deno te the torus. The F ourier coef ficients b X ( θ ) for θ ∈ T A ar e given by b X ( θ ) = E X [ e 2 π i h X , θ i ] , wher e h X , θ i = ∑ a ∈ A X a θ a . The pr obability that X = λ for λ ∈ Z A is given by th e F ourier in version formula P [ X = λ ] = Z θ ∈ T A b X ( θ ) e − 2 π i h λ , θ i d θ . Recall that our goal is to unders tand the probability that X = E [ X ] . Applying the Fourier in v ersion formula for λ = E [ X ] gi ves P [ X = E [ X ]] = Z θ ∈ T A b X ( θ ) e − 2 π i h E [ X ] , θ i d θ . (11) Thus, our goal from no w on is to understan d the Fourier coef ficients of X . W e first giv e an explic it formula for the Fourier coe ffici ents. Claim 4.2. W e have b X ( θ ) = ∏ b ∈ B ( 1 − p + pe 2 π i ·h φ ( b ) , θ i ) . Pr oof. B y definition X = φ ( T ) = ∑ b ∈ B T b φ ( b ) , wher e T b ∈ { 0 , 1 } are inde penden t w ith P [ T b = 1 ] = p . Thus b X ( θ ) = E X [ e 2 π i h X , θ i ] = E { T b : b ∈ B } [ e 2 π i ∑ b ∈ B T b h φ ( b ) , θ i ] = ∏ b ∈ B E T b [ e 2 π i T b h φ ( b ) , θ i ] = ∏ b ∈ B ( 1 − p + pe 2 π i h φ ( b ) , θ i ) . Clearly all Fourier coef ficients of X ha ve absolu te v alue at most 1. T he first step is to understan d the maximal Fourie r coef ficients of X , that is θ for which | b X ( θ ) | = 1. Claim 4.3. Let L : = { θ ∈ T A : b X ( θ ) = 1 } . Then • If θ / ∈ L then | b X ( θ ) | < 1 . • If θ ∈ L , θ ′ ∈ T A then b X ( θ + θ ′ ) = b X ( θ ′ ) . In particul ar , L is a subgr oup of T A . Pr oof. B oth claims follo w immediately from the observ ation that θ ∈ L if f h φ ( b ) , θ i ∈ Z for all b ∈ B . 12 In fact, the isolation conditions in Theorem 2.1 imply that L is a discre te subgrou p of T A (i.e. a lattice). Let M : = ( 1 / m · Z ) A be the lattice in T A of all elements w hose coordina tes are inte ger multi plies of 1 / m . W e sho w that L is a sublatt ice of M . Claim 4.4. L ⊆ M . Pr oof. L et θ ∈ L . W e need to show that m θ a ∈ Z for all a ∈ A . By the isolati on condition of Theorem 2.1, there exist s γ ∈ Z B such that φ ( γ ) = me a . Since θ ∈ L we hav e that h φ ( b ) , θ i ∈ Z for all b ∈ B . H ence also h φ ( γ ) , θ i ∈ Z , i.e. m θ a ∈ Z as claimed. The first step we ta ke is t o approx imate the Fourie r coef fi cients of X near the lat tice L . This will as sume ver y little about φ , essentially only bounded ness. The second (and more comple x) step w ill be to show that all other Fourier coefficie nts are negl igible, and in fact the contrib ution to (11) all come from Fourier coef fi cients nea r L . The second part will hea vily utilize the symmetry of the map φ and the existe nce of many disjoi nt isolation vectors . Theorem 2.1 then follo ws by a careful setting of parameters and a routine calcul ation. Formally , we will use ℓ 2 distan ce on T A . For x ∈ T define its absolute va lue | x | = | x ( mod 1 ) | to be the m inimal absolute v alue of x modulo 1 (that is, w e take x ( mod 1 ) ∈ [ − 1 / 2 , 1 / 2 ] ). Define the distance between θ ′ , θ ′′ ∈ T A by d ( θ ′ , θ ′′ ) : = r ∑ a ∈ A | θ ′ a − θ ′′ a | 2 . The distan ce between θ ∈ T A and L ⊂ T A is gi v en by d ( θ , L ) : = min α ∈ L d ( θ , α ) . The follo wing three lemmas are the main technical ingredient s of the proof. The first lemma gi ves a good approxi mation for the Fouri er coef ficients of X near zero (and by Claim 4.3, near any point in L ). Lemma 4.5 (Estimating Fourier coef ficients near zero) . Assume the conditi ons of Theor em 2.1 and fix ε ≤ O ( 1 / ( c 1 N 1 / 3 )) . Let θ ∈ T A be such tha t k θ k 2 ≤ ε . Then b X ( θ ) = e 2 π i h E [ X ] , θ i e − 4 π 2 p · θ T R θ ( 1 + δ ) wher e R is the A × A pairwise-cor r elation matrix of φ given by R a ′ , a ′′ = ∑ b ∈ B φ ( b ) a ′ φ ( b ) a ′′ , and wher e | δ | = O ( N 2 / | B | + N c 3 1 ε 3 ) . The secon d lemma bounds the Fourier coe f ficients of X far from the lattice M . Lemma 4 .6 (Bou nding Fou rier coef ficients far f rom M) . Assume t he co nditio ns of The or em 2.1. Let θ ∈ T A be such tha t d ( θ , M ) ≥ ε . Then | b X ( θ ) | ≤ exp  − N ε 2 · m 2 | A | c 2 c 2 3  . The third lemma bound s the remainin g Fourie r coefficie nts which are near M b ut far from L . In the follo wing let M \ L denote the set of elements in M b ut not in L . Lemma 4.7 (Bounding Fourier coef fi cients near M b ut far from L ) . Assume the conditions of Theor em 2.1 and fix ε ≤ 1 / ( 2 c 1 m ) . Let θ ∈ T A be such tha t d ( θ , M \ L ) ≤ ε . Then | b X ( θ ) | ≤ exp  − N · O ( 1 ) m 2 | A | log ( c 1 | A | )  . W e pr ov e Lemmas 4.5, 4.6 and 4 .7 in Sections 4.1, 4.2 an d 4.3, res pecti vely . W e comb ine them to pro ve Theorem 2.1 in Section 4.4. 13 4.1 Estimating Four ier coefficients near zer o Let θ ∈ T A be su ch that k θ k 2 ≤ ε . W e may assume that ε ≤ O ( 1 / ( c 1 N 1 / 3 )) otherwise the conc lusion of the lemma is tri vial. W e decompose e − 2 π i h E [ X ] , θ i · b X ( θ ) = ∏ b ∈ B  e − 2 π i · p h φ ( b ) , θ i · ( 1 − p + pe 2 π i h φ ( b ) , θ i )  . (12) Let ν b : = h φ ( b ) , θ i where the inner produ ct is take n over R . Since we assu me k θ k 2 ≤ ε we can bound | ν b | ≤ k φ ( b ) k 2 k θ k 2 ≤ c 1 ε ≪ 1. Thus we can appro ximate the terms in (12) by their T aylor series. The follo wing claim giv es a cubic appr oximatio n. Claim 4.8. Let f : R → C be given by f ( x ) : = e − i px ( 1 − p + pe ix ) . Then for | x | ≤ 1 we have f ( x ) = e − px 2 ( 1 + δ ) , wher e | δ | ≤ O ( p 2 x 2 + px 3 ) . Pr oof. W e compute the cubic approximatio n for f ( x ) as a polyno mial in p , x . In the follo w ing we use shorth and expressio n x = y + O ( z ) for | x − y | = O ( z ) . W e hav e f ( x ) = ( 1 − p ) e − i px + pe i ( 1 − p ) x = ( 1 − p )( 1 − i px + O ( p 2 x 2 )) + p ( 1 + i ( 1 − p ) x − x 2 ± O ( px 2 + x 3 )) = 1 − px 2 + O ( p 2 x 2 + px 3 ) = e − px 2 + O ( p 2 x 2 + px 3 ) . W e next apply the appro ximation gi ve n in Claim 4.8 to each of the terms appearing in (12). S umming up the error s, and using the fa ct that each term is bounded in absolute value by 1, we get tha t b X ( θ ) = e 2 π i h E [ X ] , θ i e − 4 π 2 p · ∑ b ∈ B ν 2 b ( 1 + δ ) (13) where | δ | ≤ O ( p 2 ∑ b ∈ B ν 2 b + p ∑ b ∈ B ν 3 b ) . T o conclude the proof, note that ∑ b ∈ B ν 2 b = ∑ b ∈ B h φ ( b ) , θ i 2 = θ T R θ , where we recall that R a ′ , a ′′ = ∑ b ∈ B φ ( b ) a ′ φ ( b ) a ′′ . T o bound the error term, recall that | ν b | ≤ c 1 ε ≪ 1 hence | δ | ≤ O ( p 2 | B | + p | B | ( c 1 ε ) 3 ) = O ( N 2 / | B | + N c 3 1 ε 3 ) . 4.2 Bounding F ourier coefficients far fr om M Let θ ∈ T A be such that d ( θ , M ) ≥ ε . Thus, there exists at least on coordin ate θ a whose distance from multiple s of 1 / m is at least ε / p | A | . Otherwise put, there exis ts a ∈ A such that | m θ a ( mod 1 ) | ≥ ε m / p | A | . (14) Recall that the Fourie r coef ficient b X ( θ ) is giv en by b X ( θ ) = ∏ b ∈ B ( 1 − p + pe 2 π i h φ ( b ) , θ i ) . 14 Hence, to get a bound on | b X ( θ ) | essentia lly w e need to sho w that h φ ( b ) , θ i is far fro m inte ger for many b ∈ B . Note that w e cannot longer assume, as in the proof of Lemma 4.5 , that h φ ( b ) , θ i is small in absolute v alue, since we assume no uppe r bo und on k θ k 2 . Thus, it may be the case that h φ ( b ) , θ i is large bu t still approx imately integer . Let ν b : = h φ ( b ) , θ i ( mod 1 ) where | ν b | ≤ 1 / 2. Our goal is to show that | ν b | is notice ably large for many v alues b ∈ B . This will then imply the require d upper bound on | b X ( θ ) | . W e will sho w this using the iso lation v ectors guarante ed by Theorem 2.1. Let γ ∈ Z B be an isolation vec tor for a with modulus m ; that is φ ( γ ) = m · e a . W e first show that it cannot be that ν b ≈ 0 for all b ∈ Supp ( γ ) . Claim 4.9. Let γ ∈ Z B be suc h that φ ( γ ) = m · e a . Then ∑ b ∈ Supp ( γ ) | ν b | 2 ≥ ε 2 m 2 | A |k γ k 2 2 . Pr oof. U sing the isolati on proper ty of γ we get that ∑ b ∈ Supp ( γ ) γ b ν b ( mod 1 ) = ∑ b ∈ Supp ( γ ) γ b h φ ( b ) , θ i ( mod 1 ) = h φ ( γ ) , θ i ( mod 1 ) = m θ a ( mod 1 ) . Hence by (14) we get that | ∑ b ∈ Supp ( γ ) γ b ν b ( mod 1 ) | ≥ ε m / p | A | . On the other hand, we can boun d | ∑ b ∈ Supp ( γ ) γ b ν b ( mod 1 ) | ≤ | ∑ b ∈ Supp ( γ ) γ b ν b | ≤ k γ k 2 s ∑ b ∈ Supp ( γ ) | ν b | 2 . Combining the two bou nds, we get that ∑ b ∈ Supp γ | ν b | 2 ≥ ε 2 m 2 / | A |k γ k 2 2 as claimed. W e no w use the assumpt ion of Theorem 2.1 on the ex istence of many vect ors which isolate a with disjoi nt support . Recall that by assumption we hav e r ≥ | B | / c 2 vec tors γ 1 , . . . , γ r ∈ Z B such that: (1) each γ i isolate s a with m odulus m ; (2) The vectors γ 1 , . . . , γ r ha ve disjoi nt supports; and (3) k γ i k ≤ c 3 for all i ∈ [ r ] . Applying Claim 4.9 to each ve ctor γ i indepe ndentl y we deriv e that ∑ b ∈ B | ν b | 2 ≥ ε 2 | B | · m 2 | A | c 2 c 2 3 . (15) T o conclude the proof of the lemma, we apply (15 ) to deri ve an upper bound on | b X ( θ ) | . The follo wing claim is simple. Claim 4.10. Let p ≤ 1 / 2 and | x | ≤ 1 / 2 . Then | 1 − p + pe 2 π ix | ≤ exp ( − px 2 ) . Applying Claim 4.10 we deri v e the bound | b X ( θ ) | = ∏ b ∈ B | 1 − p + pe 2 π i · ν b | ≤ exp − p ∑ b ∈ B | ν b | 2 ! ≤ exp  − ε 2 N · cm 2 | A | c 2 c 2 3  . 15 4.3 Bounding F ourier coefficients near M but far fr o m L Let θ ∈ T A be such that d ( θ , M \ L ) ≤ ε . That is, there ex ists α ∈ M \ L such that d ( θ , α ) ≤ ε . Since α / ∈ L there must exist b ∗ ∈ B such that h φ ( b ∗ ) , α i / ∈ Z . W e will sho w using the symmetr y of φ that in fact this holds for many b ∈ B . Moreov er , since α ∈ M we ha ve that if h φ ( b ) , α i / ∈ Z is m ust be at least 1 / m fa r from the integers . This will allow us to giv e strong upper bounds on the Fourie r coefficien t b X ( α ) and by contin uity also on b X ( θ ) . Let L denot e the lattice generated by { φ ( b ) : b ∈ B } . In other words, L is the subgroup of Z A whose elements are all possible intege r combinations of { φ ( b ) : b ∈ B } . W e first sho w that any subset of B which genera tes the lattice L must contain b for which h φ ( b ) , m α i 6 = 0. Claim 4.11. L et K ⊂ B be a set which gener ates the lattice L . Then ther e must exis t b ∈ K for which h φ ( b ) , m α i 6 = 0 . Pr oof. B y assumption since K generates the lattice L , we can exp ress φ ( b ∗ ) as an inte ger combination of { φ ( b ) : b ∈ K } . T hat is, there exi st intege r coefficien t α b for b ∈ K such that φ ( b ∗ ) = ∑ b ∈ K α b φ ( b ) . Thus, as h φ ( b ∗ ) , m α i 6 = 0, there must exi st b ∈ K for which h φ ( b ) , m α i 6 = 0 as well. W e nex t claim that there m ust exis t at least one small set K ⊂ B which gene rates L . W e will later use symmetry to genera te from it many suc h sets. Claim 4.12. Ther e e xists K ⊂ B of size | K | ≤ O ( | A | log ( c 1 | A | )) such that { φ ( b ) : b ∈ K } gen era tes th e lattice L . Pr oof. L et K be a minimal subset of B such that { φ ( b ) : b ∈ K } gene rates the lattice L . W e claim that the minimality of K implies that all partial sums φ ( K ′ ) for K ′ ⊆ K must be distinct. Otherwise , assume that there ex ist two d istinct subset s K 1 , K 2 ⊆ K for which φ ( K 1 ) = φ ( K 2 ) . W e can assume w .l.o.g that K 1 , K 2 are disjoi nt by remov ing common elements from both. Thus we hav e ∑ b ∈ K 1 φ ( b ) − ∑ b ∈ K 2 φ ( b ) = 0 . In particular , we can expr ess any b ′ ∈ K 1 ∪ K 2 as an inte ger combin ation of { φ ( b ) : b ∈ K \ { b ′ }} . Thus, we can remov e b ′ from K and maintain the propert y that the resulting set generates L . This con tradict s the minimality of K . W e thus know that all sums { φ ( K ′ ) : K ′ ⊆ K } are distinct . W e no w apply the assumption that φ is bound ed. By the assumption s of Theorem 2.1 we know that k φ ( b ) k ∞ ≤ k φ ( b ) k 2 ≤ c 1 . Hence w e conclude that { φ ( K ′ ) : K ′ ⊆ K } ⊆ [ − c 1 K , c 1 K ] A , which imply that 2 K ≤ ( 2 c 1 K + 1 ) | A | . It is easy to ve rify that this giv es the bound K ≤ O ( | A | log ( c 1 | A | )) as clai med. The nex t step is to use the symmetry of φ to gene rate m any s mall sets which span L . Claim 4.13. Let K ⊂ B be a set such that { φ ( b ) : b ∈ K } gen era tes the lattice L . Let ( π , τ ) ∈ S B × GL ( V ) be a s ymmetry of φ . Let K π : = { π ( b ) : b ∈ K } be a shift of K by π . Then { φ ( b ) : b ∈ K π } also g ener ates the lattice L . 16 Pr oof. L et b ′ ∈ B . W e n eed to sho w that we ca n e xpres s φ ( b ′ ) as inte ger combinat ion of { φ ( π ( b )) : b ∈ K } . Consider π − 1 ( b ′ ) . By assumption the image of φ on eleme nts of K genera tes the lattice L , henc e there exi st coef ficients α b ∈ Z for b ∈ K such that φ ( π − 1 ( b ′ )) = ∑ b ∈ K α b φ ( b ) . Applying the assumpti on that ( π , τ ) is a symmetry of φ we get that φ ( b ′ ) = φ ( π ( π − 1 ( b ′ ))) = τ ( φ ( π − 1 ( b ′ ))) = ∑ b ∈ K α b · τ ( φ ( b )) = ∑ b ∈ K α b φ ( π ( b )) . W e combine Claims 4.11 , 4.12 and 4.13 to deriv e that h φ ( b ) , α i 6 = 0 for many b ∈ B . Let e B = { b ∈ B : h φ ( b ) , α i 6 = 0 } . Cor ollary 4.14. | e B | ≥ Ω  | B | | A | log ( c 1 | A | )  . Pr oof. L et K be the set guar anteed by Claim 4.12 where | K | ≤ O ( | A | log ( c 1 | A | )) . Let K π = { π ( b ) : b ∈ K } . W e kno w by Claim 4.13 that for any symmet ry ( π , τ ) of φ we ha ve | K π ∩ e B | ≥ 1 . Let H be the subgroup of permutations on B gi ven by symmetries of φ . That is, H = { π : ( π , τ ) symmetry of φ } . W e kno w by the assumpt ions of Theorem 2.1 that H acts trans iti ve ly on B . Thus, for any fix ed b ∈ B , if we choo se π ∈ H uniformly we ha ve that π ( b ) is uniformly distrib uted in B . Thus, E π ∈ H [ K π ∩ e B ] = ∑ b ∈ K P π ∈ H [ π ( b ) ∈ e B ] = | K || e B | | B | . W e thus conclu de that we must ha ve | e B | ≥ | B | / | K | . W e conclu de the proof of L emma 4.7 by establishin g an upper bound of b X ( α ) . For any b ∈ e B we ha ve that h φ ( b ) , α i ( mod 1 ) 6 = 0, hence since α ∈ ( 1 / m · Z ) A we ha ve |h φ ( b ) , α i ( mod 1 ) | ≥ 1 / m . Recall that by assump tion k φ ( b ) k 2 ≤ c 1 and k α − θ k ≤ 1 / ( 2 c 1 m ) . Thus |h φ ( b ) , α − θ i| ≤ 1 / 2 m by Cauchy- Schwarz and we get tha t |h φ ( b ) , θ i ( mod 1 ) | ≥ 1 / 2 m . W e thus conclu de with an upper boun d on | b X ( θ ) | . A pplying Claim 4.10 we ha ve | b X ( θ ) | ≤ ∏ b ∈ e B | 1 − p + pe 2 π i h φ ( b ) , θ i | ≤ exp ( − p ( 1 / 2 m ) 2 | e B | ) ≤ exp  − N · O ( 1 ) m 2 | A | log ( c 1 | A | )  . 4.4 Pr oof of Theor em 2.1 from Lemmas 4.5, 4.6 and 4.7 W e no w deduce Theorem 2.1 from Lemmas 4.5, 4.6 and 4.7. Recall that we ha ve P [ X = E [ X ]] = Z θ ∈ T A b X ( θ ) e − 2 π i h E [ X ] , θ i d θ . (16) Let N = poly ( | A | , m , c 0 , c 1 , c 2 , c 3 ) lar ge enoug h to be chosen later . W e would assume throu ghout tha t N is a multiple of c 0 m . If | B | = O ( N 2 ) then the set B is small to begin w ith, so assume that | B | ≫ N 2 . W e set 17 ε ≈ N − 1 / 3 so th at the con dition s for Lemmas 4.5 and 4 .7 hol d. More expl icitly , we set ε : = O ( 1 / c 1 N 1 / 3 ) so that the co nditio ns for Lemma 4.5 h old with | δ | ≤ 1 / 2; and we assume that N ≥ Ω ( m 3 ) so that ε ≤ 1 / ( 2 c 1 m ) and the condit ions for L emma 4.7 also hold. W e decompose the integra l in (16) into three inte grals: ov er points which are ε close to L ; ove r points which a re ε close to M \ L ; and ov er points whic h are ε far from M . Our cho ice of ε < 1 / 2 m also guara ntees that balls of radius ε around distinct points in M are disjoint. W e thus ha ve that P [ X = E [ X ]] = I 1 + I 2 + I 3 where I 1 : = ∑ α ∈ L Z θ ∈ T A : d ( θ , α ) ≤ ε b X ( θ ) e − 2 π i h E [ X ] , θ i d θ , I 2 : = ∑ α ∈ M \ L Z θ ∈ T A : d ( θ , α ) ≤ ε b X ( θ ) e − 2 π i h E [ X ] , θ i d θ , I 3 : = Z θ ∈ T A : d ( θ , M ) > ε b X ( θ ) e − 2 π i h E [ X ] , θ i d θ . W e first lo wer bound I 1 . Claim 4.15. I 1 ≥ | L |  Ω ( 1 ) c 1 N 1 / 2 | A | 1 / 2  | A | . Pr oof. W e first use the assumptio n that N divid es c 0 m to reduce comput ing I 1 to an integra l around 0. W e claim that the assumpti on that c 0 m | N implies that h E [ X ] , α i ∈ Z for all α ∈ L . This is since this choice implies that all entrie s of E [ X ] are di visibl e by m since E [ X ] = N | B | φ ( B ) = ( N / c 0 m ) · m · c 0 | B | φ ( B ) ∈ m Z A . Moreo ver , since α ∈ L ⊂ M we hav e that m α ∈ Z A , hence h E [ X ] , α i ∈ Z . Combining this with Claim 4.3 which states that the Fourier coe ffici ents of X are in varia nts to shifts by α ∈ L , we deduce that I 1 = | L | Z θ ∈ T A : k θ k 2 ≤ ε b X ( θ ) e − 2 π i h E [ X ] , θ i d θ . Recall that by Lemma 4.5 and our choic e of paramete rs, if k θ k 2 ≤ ε then b X ( θ ) = e X ( θ )( 1 + δ ( θ )) where e X ( θ ) = e 2 π i h E [ X ] , θ i e − 4 π 2 p · θ T R θ and where | δ ( θ ) | ≤ 1 / 2. Hence I 1 = | L | Z θ ∈ T A : k θ k 2 ≤ ε e − 4 π 2 p · θ T R θ ( 1 + δ ) d θ . Consider I ′ 1 = | L | Z θ ∈ T A : k θ k 2 ≤ ε e − 4 π 2 p · θ T R θ d θ . W e claim that | I 1 | ≥ | I ′ 1 | / 2, hence it suf fices to lo w er boun d | I ′ 1 | in order to lo wer bo und | I 1 | . T o see that, note that I ′ 1 is an integral of a real positi ve functio n; that w e can alw ays lower bound | I 1 | by its real part Re ( I 1 ) ; and that Re ( 1 + δ ) ≥ 1 / 2 since | δ | ≤ 1 / 2. Thus | I 1 | ≥ Re ( I 1 ) ≥ Re ( I ′ 1 ) / 2 = I ′ 1 / 2 . 18 W e next lo wer bound I ′ 1 . Note first that we can bound θ T R θ ≤ B c 2 1 k θ k 2 2 . This is becau se θ T R θ = ∑ b ∈ B h θ , φ ( b ) i 2 ≤ ∑ b ∈ B k θ k 2 2 k φ ( b ) k 2 2 ≤ Bc 2 1 k θ k 2 2 . Thus we get that I ′ 1 ≥ | L | Z θ ∈ T A : k θ k 2 ≤ ε e − 4 π 2 c 2 1 N k θ k 2 2 d θ . W e bound I ′ 1 from below by the vo lume of the region in which the inte gran d is constan t. This occurs whene ver k θ k 2 ≤ ε ′ = O ( 1 / ( c 1 N 1 / 2 )) . Recall that we chose ε = O ( 1 / ( c 1 N 1 / 3 )) ≫ ε ′ . Hence the ball of radius ε ′ is conta ined in the area over which we inte grate , so we obtain the lo wer boun d | I 1 | ≥ I ′ 1 / 2 ≥ | L | · O ( 1 ) · V ol ( Ball ( 0 , ε ′ )) = L ·  Ω ( 1 ) ε ′ | A | 1 / 2  | A | = L ·  Ω ( 1 ) c 1 N 1 / 2 | A | 1 / 2  | A | . The nex t steps are to b ound I 2 and I 3 from abo ve . W e bou nd them by the maximal v alue that | b X ( θ ) | can achie ve in their integral domains. Lemma 4.7 gi ves a bound on I 2 , | I 2 | ≤ max {| b X ( θ ) | : d ( θ , M \ L ) ≤ ε } ≤ ex p  − N · O ( 1 ) m 2 | A | log ( c 1 | A | )  , and Lemma 4.6 and our choice of ε = O ( 1 / ( c 1 N 1 / 3 )) gi ves a bou nd on I 3 , | I 3 | ≤ max {| b X ( θ ) | : d ( θ , M ) ≥ ε } ≤ ex p  − N ε 2 · m 2 | A | c 2 c 2 3  = exp  − O ( N 1 / 3 ) · m 2 | A | c 2 1 c 2 c 2 3  . W e no w need to choose N lar ge enough so that I 1 ≫ | I 2 | , | I 3 | . This can be accomplis hed since I 1 decays polyn omially with N , while | I 2 | , | I 3 | decay exponen tially fa st. It is not hard to verify that this is guaranteed whene ver N ≥ Ω ( 1 ) · max ( A 2 m 2 log 2 ( mAc 0 c 1 c 2 c 3 ) , A 6 c 6 1 c 3 2 c 6 3 log 3 ( mAc 0 c 1 c 2 c 3 )) . 5 Summary and open pr oblems Our m ain theorem guaran tees the ex istenc e of a small subs et T ⊂ B for which (1) holds. T he conditions we req uire are boundedne ss, di visibility , symmetry and isola tion. The first three conditi ons seem natural for this type of problems, b ut the fourth seems artificial, as it depends on the spec ific basis we choose for V . Thus, we wonder if this condition can be removed . In particular , the followin g questi on captu res much of the dif ficulty . Let G be a group tha t acts tran siti v ely on a set X . A subset T ⊂ G is X -uni form (or an X -design ) if it acts on X exactl y as G does. T hat is, for an y x , y ∈ X , 1 | T | |{ g ∈ T : g ( x ) = y }| = 1 | G | |{ g ∈ G : g ( x ) = y }| = 1 | X | . In our languag e we may take B = G and V to be the space spanned by all functions φ ( x , y ) : B → { 0 , 1 } of the for m φ ( x , y ) ( b ) = 1 { b ( x )= y } for x , y ∈ X . Then T is X -uniform if and only if (1) holds. T aking A to be some sub set of X × X for which ( φ a ) a ∈ A forms a basis of V , the boundednes s, di visi bility and symmetry condit ions are clearly satisfied. Howe ver , it is not clear whether the isolation condition is satisfied as well. If indee d the isolation condition is redund ant, one may conjec ture that: Conjectur e 5.1. Let G be a gr oup that acts transiti vely on a set X . Then ther e exists an X -unifor m subse t T ⊂ G such tha t | T | ≤ | X | c for some unive rsa l constant c > 0 . 19 A second question is whether one can appl y our techniques to get minimal objects. Recall that the size of the objects we achiev e is only m inimal up to polynomial factors. For e xample, one of the main open proble ms in design theory is wheth er there exists a Steiner system (i.e. a t -design with λ = 1) for any t > 5. Another major open problem of a similar spirit is the exist ence of Hadamard matrices of all orders n = 4 m , or equiv alently , 2- ( 4 m − 1 , 2 m − 1 , m − 1 ) designs. Empirical estimates for n ≤ 32 sugg est that there are exp ( O ( n ( lo g n ))) Hadamard matrices of order n = 4 m . Since are so many of them, and since the logarithm of their number gro ws at a regul ar rate, we suspect that they exis t for some purely statistical reason . Howe ver , the Gaussi an local limit model seems to be fals e for H adamard matrices inte rpreted as t -designs; it does not accurately estimate how man y there are. A third questio n is whether there exists an algo rithmic version of our work, similar to the algorithmic Moser [Mos0 9] and Moser -T ardos [MT10] ver sions of the Lo v ´ asz local lemma [EL75], and the alg orithmic Bansal [Ban10] ver sion of the six stand ard d e viatio ns meth od of S pencer [Spe85]. If an ef ficient randomized algori thm of our method were found, then we could no longer indisp utably claim that we hav e a lo w- probab ility ve rsion of the probabilis tic method. On the other hand it would be strange, from the vie w point of computa tional comple xity theory , if lo w-probability existence can always be con verted to high- probab ility exi stence. Maybe our constructi on is fundamenta lly a low-p robabi lity constructio n. Refer ences [AL11] Noga Alon and Shachar Lovett, Almost k-wise vs k-wise indepen dent permu tations , and unifor - mity for gen era l gr oup actions , 2011, ECCC TR11-049. [A V97] N ogal Alon and V an H. V u, A nti-Hadamar d matrice s, coin weighing , thr eshold gates and inde- composa ble hyper gra phs , J. Combin. Theory Ser . A 79 (1997) , no. 1, 133–1 60. [Ban10] Nikhil Bansal, Constr uctive algor ithms for discr epan cy minimization , Proceedings of the 201 0 IEEE 51st Annual Symposium o n Founda tions o f Compu ter Scie nce, FOCS ’10, IEEE Computer Society , 2010, arXiv :1002.2 259, pp. 3–10. [Cam95] P . J. Cameron, P ermutatio n gr oups , Handbook of combinator ics, V ol. 1, 2, Elsevi er , 1995, pp. 611– 645. [CD07] Charles J. Colbourn and Jeff rey H . Dinitz (eds.), The CR C handbook of combinator ial desig ns , 2nd ed., Discrete Mathematics and its Applicati ons, Chapman & Hall/CRC, 2007 . [EL75] P aul Erd ˝ os and L ´ aszl ´ o L ov ´ asz, Pr oblems and r esults on 3-chr omatic hype r graphs and some re - lated ques tions , Infinite and Finite Sets, Coll. Math. Soc. J. Bolyai, no. 11, North-Holland , 1975, pp. 609– 627. [HSS99] A. S. Hedayat, N. J. A . S loane, and John S tufk en, Orthogon al arr ays: Theory and applicati ons , Springer -V erlag, 1999. [Kas07] M. K assabo v , Symmetric gr oups and expan der s , In vent. Math . 170 (200 7), no. 2, 327 –354, arXi v: math/050 3204. [KM94] Daphne K oller and Nimrod Megi ddo, Constructing small sample spaces satisfying given con- stants , S IAM J. Discrete Math. 7 (199 4), no. 2, 260–27 4. [KNR05] E. K aplan, M. Naor , and O. Reingold, Derando mized constr uction s of k -wise (almost) indepen - dent permuta tions , Approx imation, randomiz ation and combinatoria l optimizatio n (C. Chekuri, 20 K. Jansen, J. D . P . Rolim, and L . Tre visan, eds.), Lecture N otes in Computer S cience, vo l. 3624, Springer , 2005, pp. 354–365 . [KP82] Richard M. Karp and C hristos H . Papadimitriou , On linea r char acteri zation s of combinato rial optimiza tion pr oble ms , SIAM J. Comput. 11 (1982), no. 4, 620–63 2. [Mag09] Spyros S . Magli vera s, Lar ge sets of t -desi gns f r om g r oups , Math ematica Slov aca 59 (2009), no. 1, 1–20. [Mos09] Robin A. Moser , A constructiv e pr oof of the Lov ´ asz local lemma , P roceed ings of the 41st annual A CM symposium on T heory of comp uting, S TOC, A C M, 2009, arXi v:08 10.4812 , pp. 343 –350. [MT10] Robin A. Moser and G ´ abor T ardos, A constr uctive pr oof of the gener al Lov ´ asz local lemma , J. A CM 57 (2010), no. 2, 11:1–11:15 , arXiv: 0903.05 44. [Rao73] C. Radhakrishna Rao, Some combinato rial pr oblems of arr ays and application s to desig n of ex- periment s , Surve y of combin atorial theory (J. N . Sri v asta va, ed.), North-Holland , 1973, pp. 349– 359. [Spe85] Joel Spence r , Six sta ndar d devia tions suf fice , Tran s. Amer . Math. Soc. 289 (19 85), no. 2, 679 – 706. [T ei87] Luc T eirlinck, Non-trivi al t -designs without r epeate d bloc ks e xist for all t , Discrete Math. 65 (1987 ), no. 3, 301–3 11. 21