Lower bounds for measurable chromatic numbers

LO WER BOUNDS FOR ME ASURAB LE CHR OMA TIC NUMBERS CHRISTINE B ACHOC, GABRIELE NEBE, FERN ANDO M ´ ARIO DE OLIVEIRA FILHO, AND FRANK V ALLENTIN A B S T R A C T . The Lo v ´ asz theta function provid es a lo wer bound for the chro- matic number of finite graph s based on the solution of a semidefinite program. In this paper we gene ralize it so that it gi ves a lo wer bound for the measurab l e chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. T here we trans- form the original infinite semidefinite program into an infinite linear program which then turns out to be an ex tremal question ab out Jacobi polyno mi als which we solv e explicitly in the li mit. As an ap plication w e deriv e ne w lo wer bounds for the measurable chromatic number of the Euclidean space i n dimensions 10 , . . . , 24 and we giv e a new proof that it gro ws expo nentially wit h the dimension . 1. I N T R O D U C TI O N The chr omati c number of the n -dimensio nal E uclide an space is the minimum number of colo rs needed to color each point of R n in such a way that points at distan ce 1 from each other rece ive diffe rent colors. It is the chromati c number of the g raph with v erte x set R n and in which tw o v ertices are adjace nt if their distance is 1 . W e deno te it by χ ( R n ) . A famo us open question is to determine the chromat ic number of the plane. In this case, it is only known that 4 ≤ χ ( R 2 ) ≤ 7 , w here lo wer and upper bounds come from simple geometri c const ructio ns. In this form the pro blem was consid- ered, e.g., by Nelson, Isbell, Erd ˝ os, and H adwiger . For historical remarks and for the best kno wn bound s in other dimensions we refer to Sz ´ ekely’ s surve y article [21]. The fi rst e xponential asymptotic lo wer bound is du e to Frankl and W ilson [8, Theorem 3]. Currently the best kno w n asymptotic lo w er bound is due to R aig- orodsk ii [17] and the best known asymptotic uppe r bound is due to Larman and Rogers [12]: (1 . 239 . . . + o (1)) n ≤ χ ( R n ) ≤ (3 + o (1)) n . In this paper we study a vari ant of the chromatic number of R n , namely the measurab le chromat ic number . The measura ble chr omatic number of R n is the smallest number m such that R n can be parti tioned into m Lebesgue measu rable Date : July 17, 2009. 1991 Mathematics Subject Classification. 52C10, 52C17, 90C22. K ey wor ds and phrases. Nel son-Hadwiger problem, measurable chromatic number , semidefinite programming, orthogona l polynomials, spherical codes. The third author was partially supported by CA PES/Brazil under grant BEX 2421/04-6. The fourth author was partially supported by the D eutsche Forschungsgem einschaft (DFG) under grant SCHU 1503/4. 1 2 C. Bachoc, G. Nebe, F .M. de Oli veira Filho, F . V allenti n stable s ets. Here we call a s et C ⊆ R n stable i f no tw o points in C lie at di stance 1 from each o ther . In other wo rds, we impose that the se ts of points ha ving the same color ha ve to be measurable. W e denote the m easura ble chro m atic number of R n by χ m ( R n ) . One reason to study the m easurab le chromat ic number is that then strong er analytic tools are av ailable. The study of the measurable chro m atic number start ed with Falconer [7], who pro ved that χ m ( R 2 ) ≥ 5 . The m easura ble chromati c number is at least the chro- matic number and it is amusing to notice that in case of strict inequality the con- structi on of an optimal coloring necessarily uses the axiom of choice. Related to the chromatic number of the Euc lidean s pace is the chromatic number of the unit sphere S n − 1 = { x ∈ R n : x · x = 1 } . For − 1 < t < 1 , we consi der the graph G ( n, t ) whose vertic es are the points of S n − 1 and in which two points are adjacen t if their inner produ ct x · y equals t . The chroma tic number of G ( n, t ) and its measurable versi on, deno ted by χ ( G ( n, t )) and χ m ( G ( n, t )) respecti vely , are defined as in the Euclid ean case. The chr omatic nu mber of this graph w as stud ied by Lo v ´ asz [ 14], in particula r in the case when t is small. He sho wed that n ≤ χ ( G ( n, t )) for − 1 < t < 1 , χ ( G ( n, t )) ≤ n + 1 for − 1 < t ≤ − 1 /n. Frankl and W ilson [8, Theorem 6] sho wed that (1 + o (1))(1 . 13) n ≤ χ m ( G ( n, 0)) ≤ 2 n − 1 . The (measurable ) chromatic number of G ( n, t ) pro vides a lo wer bound for the one of R n : After appr opriat e scalin g, eve ry proper colorin g of R n interse cted w ith the unit s phere S n − 1 gi ves a prope r colorin g of the gr aph G ( n, t ) , and measurabi l- ity is preser ved by the intersec tion. In this paper we pres ent a lower bound for the measurable chromati c number of G ( n, t ) . As an applic ation we deri ve ne w lo wer bounds for the measurab le chromati c number of the Euclidean space in dimensions 10 , . . . , 24 and w e giv e a ne w proof that it grows e xponentially w ith the dimension . The lo wer bound is based o n a genera lization of the Lov ´ asz theta function (Lov ´ asz [ 13 ]), wh ich giv es an upp er bo und to the stability numb er of a finite graph. Here we aim at gene ralizin g the the ta function to distanc e gra phs in compac t m et- ric spaces. Thes e are graphs defined on all poin ts of the metric space where the adjace ncy relatio n only depends on the distance. The paper is structured as follo ws: In Section 2 we define the stability numbe r and the fractio nal measu rable chromatic n umber and giv e a basic inequalit y in v olv- ing them. Then, after revie wing Lov ´ asz’ original formulation of the theta function in Section 3, we giv e our generalizat ion in Section 4. Like the original theta func- tion for finite graphs, it gi ves an upper bound for the stability n umber . Moreove r , in the c ase o f the unit sphere, it c an be e xplicitly computed , thank s to classica l results on sphe rical harmoni cs. The materia l needed for spher ical harmonics is giv en in Section 5 and an exp licit formulatio n for the theta funct ion of G ( n, t ) is gi ven in Section 6. Lowe r bounds for measurabl e chromatic numbers 3 In Section 7 we choose specific value s of t for which w e can analytical ly com- pute the theta funct ion of G ( n, t ) . This allows us to compute the limit of the theta functi on for th e gra ph G ( n, t ) as t goes to 1 in Sectio n 8. This g i ves impro vements on the best kno wn lo wer bounds for χ m ( R n ) in se veral dimensions . Furt hermore this giv es a ne w proof of the fact that χ m ( R n ) grows exponen tially with n . Al- thoug h this is an immediate conseque nce of the result of Frankl and W ilson (and of Raigorodskii , and also of a result of Frankl and R ¨ od l [9]) and our bound of 1 . 165 n is not an impro veme nt, our result is an easy conse quence of the methods we present . Moreo ver , we think that our proof is of interest beca use the metho ds we use here are radically diff erent from those used before. In particular , they can be appl ied to other m etric space s. In Section 9 we point out ho w to apply our general izatio n to dista nce graphs in other compact metric spaces, endo wed with the contin uous actio n of a compact group . Finally in Section 10 we conclude by s ho wing the relatio n between our gen- eraliza tion of the theta function and the the ta function for finite graphs of G ( n, t ) and by sho wing the relation between our general izatio n and the linear prog ram- ming boun d for spherical codes establishe d by Delsarte, Goethals , and S eidel [6]. 2. T H E F R A C T I O N A L C H R O M A T I C N U M B E R A N D T H E S T A B I L I T Y N U M B E R Let G = ( V , E ) be a finite or infinite graph whose verte x set is equippe d with the measure µ . W e ass ume that the measure o f V is finite. In this section we define th e stabili ty number and the measurabl e fractio nal chromati c number of G and deri ve the basic inequality between these two in v ariants. In the case of a finite graph one reco vers the classical notions if one uses the uniform measure µ ( C ) = | C | for C ⊆ V . Let L 2 ( V ) be the Hilbert space of real-v alued square -inte grable functions de- fined ov er V with inner product ( f , g ) = Z V f ( x ) g ( x ) dµ ( x ) for f , g ∈ L 2 ( V ) . T he constant functio n 1 is measurable and its squared norm is the number (1 , 1) = µ ( V ) . The chara cterist ic funct ion of a subset C of V we denote by χ C : V → { 0 , 1 } . A su bset C of V is cal led a measur able stable set if C is a measura ble set an d if no two v ertices in C are adjacent. The stability number of G is α ( G ) = sup { µ ( C ) : C ⊆ V is a measur able stable set } . Similar measure-theo retical notions of the stabili ty number hav e been considered before by othe r autho rs for the case in which V is the Euclidean spac e R n or the sphere S n − 1 . W e refer the reader to the surve y paper of Sz ´ ekely [21] for more informat ion and further references. The fractiona l measurab le ch r omatic number of G is den oted by χ ∗ m ( G ) . It is the infimum of λ 1 + · · · + λ k where k ≥ 0 and λ 1 , . . . , λ k are nonne gativ e real numbers such that there e xist measurab le stable sets C 1 , . . . , C k satisfy ing λ 1 χ C 1 + · · · + λ k χ C k = 1 . 4 C. Bachoc, G. Nebe, F .M. de Oli veira Filho, F . V allenti n Note that the measurabl e fractiona l chromatic number of the graph G is a lo wer bound for its measura ble chromatic number . Pro p osition 2.1. W e have the followin g basic inequalit y bet ween the st ability num- ber and the measu rab le fr action al chr omatic number of a grap h G = ( V , E ) : (1) α ( G ) χ ∗ m ( G ) ≥ µ ( V ) . So, any uppe r bound for α ( G ) pr ovides a lower bound for χ ∗ m ( G ) . Pr oof. Let λ 1 , . . . , λ k be no nnegati ve real numbe rs and C 1 , . . . , C k be meas urable stable sets such that λ 1 χ C 1 + · · · + λ k χ C k = 1 . Since C i is measura ble, its char - acteris tic function χ C i lies in L 2 ( V ) . Hence ( λ 1 + · · · + λ k ) α ( G ) ≥ λ 1 µ ( C 1 ) + · · · + λ k µ ( C k ) = λ 1 ( χ C 1 , 1) + · · · + λ k ( χ C k , 1) = (1 , 1) = µ ( V ) .  3. T H E L OV ´ A SZ T H E T A F U N C T I O N F O R FI N I T E G R A P H S In the celebrat ed paper [13] Lov ´ asz introdu ced the theta function for finite graphs. It is an upper bound for the stability number which one can efficie ntly compute using semide finite programming. In this section we re view its definitio n and prop- erties, which we genera lize in Section 4. The theta functio n of a grap h G = ( V , E ) is defined by ϑ ( G ) = max  X x ∈ V X y ∈ V K ( x, y ) : K ∈ R V × V is posit i ve semidefini te , X x ∈ V K ( x, x ) = 1 , K ( x, y ) = 0 if { x, y } ∈ E  . (2) Theor em 3.1. F or any fini te gra ph G , ϑ ( G ) ≥ α ( G ) . Although this result follo ws from [13, L emma 3] and [13, Theorem 4], w e gi ve a proof here t o stress the a nalog y between the finite cas e and the mo re ge neral ca se we cons ider in our generali zation Theorem 4.1. Pr oof of Theor em 3.1 . Let C ⊆ V be a stable set . Consid er th e character istic func- tion χ C : V → { 0 , 1 } of C and define the matrix K ∈ R V × V by K ( x, y ) = 1 | C | χ C ( x ) χ C ( y ) . Notice K satisfies the conditions in (2). Moreo ver , we hav e P x ∈ V P y ∈ V K ( x, y ) = | C | , and so ϑ ( G ) ≥ | C | .  Remark 3.2. Ther e ar e many equivale nt definitions of the theta func tion. P ossible altern atives are r e viewed by Knuth in [11] . W e use the one of [13, Theorem 4] . Lowe r bounds for measurabl e chromatic numbers 5 If the graph G has a nontri vial automorphism group, it is not dif fi cult to see that one can restrict oneself in (2) to the functions K which are in v ariant under the action of any subgrou p Γ of Aut( G ) , where Aut( G ) is the automorphism gr oup of G , i.e., it is the group of all permut ations of V that preserve adjacenc y . Here we say that K is in variant under Γ if K ( γ x, γ y ) = K ( x, y ) holds for all γ ∈ Γ and all x, y ∈ V . If moreov er Γ acts trans itiv ely on G , the second condit ion P x ∈ V K ( x, x ) = 1 is equiv alent to K ( x, x ) = 1 / | V | for all x ∈ V . 4. A G E N E R A LI Z A T I O N O F T H E L OV ´ A SZ T H E TA F U N C T I O N F O R D I S T A N C E G R A P H S O N C O M P A C T M E T R I C S P AC E S W e assume that V is a compact metric space with distance function d . W e moreo ver assume that V is equipped with a nonne gativ e, Borel regular measure µ for which µ ( V ) is finite. Let D be a closed subset of the image of d . W e define the graph G ( V , D ) to be the graph with v ertex set V and edg e set E = {{ x, y } : d ( x, y ) ∈ D } . The elements of the space C ( V × V ) consisting of all continu ous functions K : V × V → R are called cont inuou s Hilbert-Sc hmidt kernels ; or ker nels for short. In the follo wing we only cons ider symmetric ke rnels, i.e., kernels K with K ( x, y ) = K ( y , x ) for all x, y ∈ V . A kern el K ∈ C ( V × V ) is called positiv e if, for any nonnegat i ve integer m , an y points x 1 , . . . , x m ∈ V , and an y real numbers u 1 , . . . , u m , we ha ve m X i =1 m X j =1 K ( x i , x j ) u i u j ≥ 0 . W e are now ready to extend the definition (2) of the Lo v ´ asz theta function to the graph G ( V , D ) . W e define ϑ ( G ( V , D )) = sup  Z V Z V K ( x, y ) dµ ( x ) dµ ( y ) : K ∈ C ( V × V ) is positi ve , Z V K ( x, x ) dµ ( x ) = 1 , K ( x, y ) = 0 if d ( x, y ) ∈ D  . (3) Theor em 4.1. The theta function is an upper bound for the stabilit y number , i.e., ϑ ( G ( V , D )) ≥ α ( G ( V , D )) . Pr oof. Fix ε > 0 arbit rarily . Let C ⊆ V be a stable set such that µ ( C ) ≥ α ( G ( V , D )) − ε . Since µ is regular , we may assume that C is closed , as oth- erwise we could find a stable set w ith measure closer to α ( G ( V , D )) and use a suitab le inner -appro ximation of it by a closed set. Note that, since C is compa ct and stable , there must exist a number β > 0 such that | d ( x, y ) − δ | > β for all x, y ∈ C and δ ∈ D . But then, for small enough ξ > 0 , the set B ( C, ξ ) = { x ∈ V : d ( x, C ) < ξ } , 6 C. Bachoc, G. Nebe, F .M. de Oli veira Filho, F . V allenti n where d ( x, C ) is the distance from x to the closed set C , is stable. M oreo ver , no tice that B ( C, ξ ) is open and that, since it is stable, µ ( B ( C, ξ )) ≤ α ( G ( V , D )) . No w , th e functi on f : V → [0 , 1] gi ven by f ( x ) = ξ − 1 · m ax { ξ − d ( x, C ) , 0 } for all x ∈ V is continuo us and such that f ( C ) = 1 and f ( V \ B ( C, ξ )) = 0 . So the k ernel K giv en by K ( x, y ) = 1 ( f , f ) f ( x ) f ( y ) for all x, y ∈ V is feasible in (3). Let us estimate the object i ve va lue of K . Since we hav e ( f , f ) ≤ µ ( B ( C, ξ )) ≤ α ( G ( V , D )) and Z V Z V f ( x ) f ( y ) dµ ( x ) dµ ( y ) ≥ µ ( C ) 2 ≥ ( α ( G ( V , D )) − ε ) 2 , we finally ha ve Z V Z V K ( x, y ) dµ ( x ) dµ ( y ) ≥ ( α ( G ( V , D )) − ε ) 2 α ( G ( V , D )) and, sinc e ε is arbitrary , the theorem follo ws.  Let us no w assume that a compact group Γ acts continuous ly on V , preserving the distance d . T hen, if K is a feasibl e solution for (3), so is ( x, y ) 7→ K ( γ x, γ y ) for all γ ∈ Γ . A veragin g on Γ leads to a Γ -in varia nt feasible solution K ( x, y ) = Z Γ K ( γ x, γ y ) dγ , where dγ denotes the Haar measure on Γ normalized so that Γ has vo lume 1 . Moreo ver , ob serv e that the object ive valu e of K is the same as that of K . H ence we can restric t oursel ves in (3) to Γ -in v ariant kernels . If moreov er V is homogeneo us under the action of Γ , the second condition in (3) m ay be replaced by K ( x, x ) = 1 /µ ( V ) for all x ∈ V . W e are mostly intereste d in the case in which V is the unit spher e S n − 1 endo wed with the Euclidean metric of R n , and in which D is a single ton. If D = { δ } and δ 2 = 2 − 2 t , so that d ( x, y ) = δ if and only if x · y = t , the graph G ( S n − 1 , D ) is denote d by G ( n, t ) . S ince the unit sphere is homogeneous under the action of the orthog onal group O( R n ) , the pre vious remarks apply . 5. H A R M O N I C A N A L Y SI S O N T H E U N I T S P H E R E It turns out tha t the continuo us posi ti ve Hilbert -Schmidt kerne ls on the sphere ha ve a nice desc ription coming from classical result s o f harmonic analys is re- vie wed in this section. This allo ws for the calcula tion of ϑ ( G ( n, t )) . For info r - mation on spheri cal harmonics w e refer to [1, Chapt er 9] and [23]. The unit spher e S n − 1 is homogene ous under the action of the orthogonal group O( R n ) = { A ∈ R n × n : A t A = I n } , w here I n denote s the identity matrix. More- ov er , it is two-poin t homogeneous , meaning that the orbits of O( R n ) on pairs of Lowe r bounds for measurabl e chromatic numbers 7 points are characterized by the v alue of their inner product. The orthogona l group acts on L 2 ( S n − 1 ) by Af ( x ) = f ( A − 1 x ) , and L 2 ( S n − 1 ) is equip ped with the standa rd O( R n ) -in v ariant inner produ ct (4) ( f , g ) = Z S n − 1 f ( x ) g ( x ) dω ( x ) for the standar d surf ace measure ω . The surface area of the unit sphere is ω n = (1 , 1) = 2 π n/ 2 / Γ( n/ 2) . It is a w ell-kno wn fact (see e.g. [23, Chapter 9.2]) that the H ilbert space L 2 ( S n − 1 ) decompo ses under the action of O( R n ) into ortho gonal subspaces (5) L 2 ( S n − 1 ) = H 0 ⊥ H 1 ⊥ H 2 ⊥ . . . , where H k is isomorphi c to the O( R n ) -irred ucible space Harm k = n f ∈ R [ x 1 , . . . , x n ] : f homogeneo us , deg f = k , n X i =1 ∂ 2 ∂ x 2 i f = 0 o of harmonic polynomia ls in n vari ables which are homogeneo us and ha ve de gree k . W e set h k = dim(Harm k ) =  n + k − 1 n − 1  −  n + k − 3 n − 1  . The equality in (5) means that ev ery f ∈ L 2 ( S n − 1 ) can be uniq uely writte n in th e form f = P ∞ k =0 p k , where p k ∈ H k , and where the con v ergenc e is in the L 2 -norm. The addition formula (see e.g. [1, Chapter 9.6]) plays a centra l role in the char- acteriz ation of O( R n ) -in v ariant kern els: For any ort honor mal basis e k , 1 , . . . , e k ,h k of H k and for any p air of points x, y ∈ S n − 1 we ha ve (6) h k X i =1 e k ,i ( x ) e k ,i ( y ) = h k ω n P ( α,α ) k ( x · y ) , where P ( α,α ) k is the normalized Jacobi polynomial of degree k with parameters ( α, α ) , with P ( α,α ) k (1) = 1 and α = ( n − 3) / 2 . The J acobi polynomials with paramete rs ( α, β ) are orth ogona l polyno m ials for the weight function (1 − u ) α (1 + u ) β on the in terv al [ − 1 , 1] . W e denote b y P ( α,β ) k the normalize d Jacobi po lynomial of de gree k with normaliza tion P ( α,β ) k (1) = 1 . In [18, T heorem 1] Schoenb er g gav e a characte rizatio n of the cont inuou s ker - nels which are positi ve and O( R n ) -in v ariant: They are those w hich lie in the cone spann ed by the k ernels ( x, y ) 7→ P ( α,α ) k ( x · y ) . More p recisely , a continu ous ker nel K ∈ C ( S n − 1 × S n − 1 ) is O( R n ) -in v ariant and positi ve if and only if there ex ist nonne gati ve real numbers f 0 , f 1 , . . . such that K can be written as (7) K ( x, y ) = ∞ X k =0 f k P ( α,α ) k ( x · y ) , where the con ver gence is absolu te and uniform. 8 C. Bachoc, G. Nebe, F .M. de Oli veira Filho, F . V allenti n 6. T H E T H E T A F U N C T I O N O F G ( n, t ) W e obtain from Section 4 in the case V = S n − 1 , D = { √ 2 − 2 t } , and Γ = O( R n ) , the follo wing characte rizatio n of the theta function of the graph G ( n, t ) = G ( S n − 1 , D ) : ϑ ( G ( n, t )) = max  Z S n − 1 Z S n − 1 K ( x, y ) dω ( x ) dω ( y ) : K ∈ C ( S n − 1 × S n − 1 ) is positi ve , K is in v ariant under O( R n ) , K ( x, x ) = 1 /ω n for all x ∈ S n − 1 , K ( x, y ) = 0 if x · y = t  . (8) (It will be clear later that the maximum abo ve indeed exists.) Cor ollary 6.1. W e have ω n /ϑ ( G ( n, t )) ≤ χ ∗ m ( G ( n, t )) . Pr oof. Immediate from Theorem 4.1 and the consider ations in Section 2.  A result of de Bruijn and E rd ˝ os [4] impli es that th e chromati c number of G ( n, t ) is attain ed by a finite subgraph of it. So one might wonder if computin g the theta functi on fo r a fi nite subgr aph o f G ( n, t ) could giv e a be tter bou nd than the p reviou s coroll ary . This is not the case as we will sho w in Section 10. The theta funct ion for finite graphs has the important property that it can be compute d in polynomial time, in the sense that it can be approxi mated with arbi- trary precisio n using semidefinite programming. W e no w turn to the probl em of computin g the generaliz ation (8). First, we apply Schoenbe rg’ s charac terization (7) of the conti nuous k ernels which are O( R n ) -in v ariant and positi ve. This transforms the original formula tion (3), which is a semidefinite programming proble m in infinitely many varia bles ha ving infinitely many constrain ts, into the follo wing linear progra mming problem w ith optimiza tion variab les f k : ϑ ( G ( n, t )) = max  ω 2 n f 0 : f k ≥ 0 for k = 0 , 1 , . . . , ∞ X k =0 f k = 1 /ω n , f 0 + ∞ X k =1 f k P ( α,α ) k ( t ) = 0  , (9) where α = ( n − 3) / 2 . T o o btain (9) we s implified the object i ve funct ion in the follo wing way . Because of the orthogona l dec omposit ion ( 5) and beca use the subspa ce H 0 contai ns o nly the Lowe r bounds for measurabl e chromatic numbers 9 consta nt functions, we hav e Z S n − 1 Z S n − 1 ∞ X k =0 f k P ( α,α ) k ( x · y ) dω ( x ) dω ( y ) = ω 2 n f 0 . W e furthe rmore used P ( α,α ) 0 = 1 and P ( α,α ) k (1) = 1 . Theor em 6.2. Let m ( t ) be the minimum of P ( α,α ) k ( t ) for k = 0 , 1 , . . . Then the optimal valu e of (9) is equal to ϑ ( G ( n, t )) = ω n m ( t ) m ( t ) − 1 . Pr oof. W e first claim that the m inimum m ( t ) exists and is nega tive . Indeed, if P ( α,α ) k ( t ) ≥ 0 for all k ≥ 1 , then (9) eithe r has no solu tion (in the case that all P ( α,α ) k ( t ) are positiv e) or f 0 = 0 in any solution, which contr adicts Theorem 4.1. So we know that for some k ≥ 1 , P ( α,α ) k ( t ) < 0 . This, combined with the fact that P ( α,α ) k ( t ) goes to zero as k goes to infinity (cf. [1, Chapter 6.6] or [2 0, Chapt er 8.22]), pro ves the clai m. Let k ∗ be so that m ( t ) = P ( α,α ) k ∗ ( t ) . I t is easy to see that there is an optimal soluti on of (9 ) in which only f 0 and f k ∗ are positiv e. H ence, solving the resulting system f 0 + f k ∗ = 1 /ω n f 0 + f k ∗ m ( t ) = 0 gi ves f 0 = m ( t ) / ( ω n ( m ( t ) − 1)) and f k ∗ = − 1 / ( ω n ( m ( t ) − 1)) and the theo rem follo ws.  Example 6.3. The minimum of P ( α,α ) k (0 . 999 9) for α = (24 − 3) / 2 is attained at k = 1131 . It is a rationa l number and its first decimal digits ar e − 0 . 00059623 . 7. A NA L Y T I C S O L U T I O N S In this section we compute the v alue m ( t ) = min { P ( α,α ) k ( t ) : k = 0 , 1 , . . . } for specific value s of t . Namely we choose t to be the larges t zero of an appropriate Jacobi polyn omial. Ke y for the discussi on to follow is the interl acing pr operty of the zeroes of orthog onal polynomials. It says (cf. [20, Theorem 3.3.2]) that between any pair of consec utiv e zeroes of P ( α,α ) k there is exac tly one zero of P ( α,α ) k − 1 . W e denote the zeros of P ( α,β ) k by t ( α,β ) k ,j with j = 1 , . . . , k and with the increa s- ing orderin g t ( α,β ) k ,j < t ( α,β ) k ,j +1 . W e shal l need the followin g collection of identities: (10) (1 − u 2 ) d 2 P ( α,α ) k du 2 − (2 α + 2) u dP ( α,α ) k du + k ( k + 2 α + 1) P ( α,α ) k = 0 , 10 C. Bachoc, G. Nebe, F .M. de Oli veira Filho, F . V allenti n ( − 1) k P ( α,α ) k ( − u ) = P ( α,α ) k ( u ) , (11) ( − 1) k ( α + 1) P ( α,α +1) k ( − u ) = ( k + α + 1) P ( α +1 ,α ) k ( u ) , (12) (2 α + 2) dP ( α,α ) k du = k ( k + 2 α + 1) P ( α +1 ,α +1) k − 1 , (13) (2 α + 2) P ( α,α +1) k = ( k + 2 α + 2) P ( α +1 ,α +1) k − k P ( α +1 ,α +1) k − 1 , (14) (2 k + 2 α + 2) P ( α +1 ,α ) k = ( k + 2 α + 2) P ( α +1 ,α +1) k + k P ( α +1 ,α +1) k − 1 , (15) ( k + α + 1) P ( α +1 ,α ) k = ( α + 1) P ( α,α ) k − P ( α,α ) k +1 1 − u . (16) They can all be found in [1, Chapter 6], althou gh with differe nt normali zation. Formula (10) is [1, (6.3.9)] ; (11) and (12) are [1, (6.4.23)]; (13) is [ 1 , (6.3.8)], (14) is [1, (6.4.21)]; (15) follo ws by the change of v ariables u 7→ − u from (14 ) and (11), (12); (16) is [1, (6.4.20)] . Pro p osition 7.1. Let t = t ( α +1 ,α +1) k − 1 ,k − 1 be the lar gest zer o of the J acobi polynomial P ( α +1 ,α +1) k − 1 . Then, m ( t ) = P ( α,α ) k ( t ) . Pr oof. W e start with the follo wing crucial o bserv ation: From (13), t is a zero of the deri vat i ve of P ( α,α ) k . Hence it is a minimum of P ( α,α ) k becaus e it is the last extre m al v alue in the interv al [ − 1 , 1] and because P ( α +1 ,α +1) k (1) = 1 , whence (using (13)) P ( α,α ) k ( u ) is increasing on [ t, 1] . No w we prov e that P ( α,α ) k ( t ) < P ( α,α ) j ( t ) for all j 6 = k w here w e t reat the cases j < k and j > k separately . It turns out that the sequence P ( α,α ) j ( t ) is decreasi ng for j ≤ k . From (16), the sign of P ( α,α ) j ( t ) − P ( α,α ) j +1 ( t ) equals the sign of P ( α +1 ,α ) j ( t ) . W e hav e the inequa lities t ( α +1 ,α ) j,j ≤ t ( α +1 ,α ) k − 1 ,k − 1 < t ( α +1 ,α +1) k − 1 ,k − 1 = t. The fi rst one is a consequen ce of the interlacing property . From (15) one can deduc e that P ( α +1 ,α ) k − 1 has exa ctly one zero in the interv al [ t ( α +1 ,α +1) k − 2 ,i − 1 , t ( α +1 ,α +1) k − 1 ,i ] since it changes sign at the ext reme points of it, and by the same ar gument P ( α +1 ,α ) k − 1 has a zero left to t ( α +1 ,α +1) k − 1 , 1 . Thus, t ( α +1 ,α ) k − 1 ,k − 1 < t ( α +1 ,α +1) k − 1 ,k − 1 = t . So t lies to the right of the large st zero of P ( α +1 ,α ) j and hence P ( α +1 ,α ) j ( t ) > 0 which sho ws that P ( α,α ) j ( t ) − P ( α,α ) j +1 ( t ) > 0 for j < k . Let us conside r the case j > k . The inequa lity [1, (6.4.19)] implies that (17) for all j > k , P ( α,α ) k ( t ( α +1 ,α +1) k − 1 ,k − 1 ) < P ( α,α ) j ( t ( α +1 ,α +1) j − 1 ,j − 1 ) . The next observ ation, which finishes the proof of the lemma, is stated in [1 , (6.4.24) ] only for the case α = 0 : (18) for all j ≥ 2 , min { P ( α,α ) j ( u ) : u ∈ [0 , 1] } = P ( α,α ) j ( t ( α +1 ,α +1) j − 1 ,j − 1 ) . Lowe r bounds for measurabl e chromatic numbers 11 T o pro ve it con sider g ( u ) = P ( α,α ) j ( u ) 2 + 1 − u 2 j ( j + 2 α + 1)  dP ( α,α ) j du  2 . Applying (10) in the compu tation of g ′ sho ws that g ′ ( u ) = (4 α + 2) u j ( j + 2 α + 1)  dP ( α,α ) j du  2 . The polynomial g ′ tak es posit i ve value s on [0 , 1] and hence g is increasi ng on this interv al. In particul ar , g ( t ( α +1 ,α +1) j − 1 ,i − 1 ) < g ( t ( α +1 ,α +1) j − 1 ,i ) fo r all i ≤ j − 1 with t ( α +1 ,α +1) j − 1 ,i − 1 ≥ 0 , which simplifies to P ( α,α ) j ( t ( α +1 ,α +1) j − 1 ,i − 1 ) 2 < P ( α,α ) j ( t ( α +1 ,α +1) j − 1 ,i ) 2 . Since t ( α +1 ,α +1) j − 1 ,i are the local ext rema of P ( α,α ) j , we hav e prove d (18).  8. N EW L OW E R B O U N D S F O R T H E E U C L I D E A N S P A C E In this section we giv e ne w lower bound s for the measurabl e chromatic number of the Euclidean space for dimens ions 10 , . . . , 24 . This improv es on the pre vious best kno w n lower bounds due to Sz ´ ekel y and W ormald [22]. T able 8.1 compares the values . Furthe rmore we gi ve a new proof that t he measurable chromat ic number gro ws expone ntially with the dimensi on. For this we gi ve a closed expres sion for lim t → 1 m ( t ) which in vo lves the Bessel functi on J α of the first kind of order α = ( n − 3) / 2 (see e.g. [1, Chapter 4]). The appearan ce of B essel function s here is due to the fact that the lar gest zero of the Jaco bi poly nomial P ( α,α ) k beha ves lik e the fi rst posi ti ve zero j α of the Bessel functi on J α . More pre cisely , it is kno wn [1, Theore m 4.14.1] that, for the larges t zero t ( α +1 ,β ) k ,k = cos θ k of the polyn omial P ( α +1 ,β ) k , (19) lim k →∞ k θ k = j α +1 and, with our normaliz ation (cf. [1, T heorem 4.11.6]) , (20) lim k →∞ P ( α,α ) k  cos u k  = 2 α Γ( α + 1) J α ( u ) u α . Theor em 8.1. W e hav e lim t → 1 m ( t ) = 2 α Γ( α + 1) J α ( j α +1 ) ( j α +1 ) α . Pr oof. First we sho w that (21) lim k →∞ P ( α,α ) k ( t ( α +1 ,β ) k − 1 ,k − 1 ) = 2 α Γ( α + 1) J α ( j α +1 ) ( j α +1 ) α . 12 C. Bachoc, G. Nebe, F .M. de Oli veira Filho, F . V allenti n W e estimate the dif ference | P ( α,α ) k ( t ( α +1 ,β ) k − 1 ,k − 1 ) − 2 α Γ( α + 1) J α ( j α +1 ) ( j α +1 ) α | , that we upp er bound by     P ( α,α ) k ( t ( α +1 ,β ) k − 1 ,k − 1 ) − P ( α,α ) k  cos j α +1 k      +     P ( α,α ) k  cos j α +1 k  − 2 α Γ( α + 1) J α ( j α +1 ) ( j α +1 ) α     . The secon d term tends to 0 from (20). Define θ k − 1 by t ( α +1 ,β ) k − 1 ,k − 1 = cos θ k − 1 . By the mean v alue theorem we hav e     P ( α,α ) k ( t ( α +1 ,β ) k − 1 ,k − 1 ) − P ( α,α ) k  cos j α +1 k      ≤  max u ∈ [ − 1 , 1]   dP ( α,α ) k du      cos θ k − 1 − cos j α +1 k   ≤  max u ∈ [ − 1 , 1]   dP ( α,α ) k du     max θ ∈ I k | sin θ |    θ k − 1 − j α +1 k   , where I k denote s the interv al with extremes θ k − 1 and j α +1 k . Then, with (19), θ k − 1 − j α +1 k = θ k − 1 − j α +1 k − 1 + j α +1 k ( k − 1) = 1 k − 1 (( k − 1) θ k − 1 − j α +1 ) + j α +1 k ( k − 1) = o  1 k  , and for all θ ∈ I k | sin θ | ≤ | θ | ≤ j α +1 k +   θ k − 1 − j α +1 k   = O  1 k  . From (13), max u ∈ [ − 1 , 1]    dP ( α,α ) k du    ∼ k 2 . Hence we ha ve prov ed that lim k →∞     P ( α,α ) k ( t ( α +1 ,β ) k − 1 ,k − 1 ) − P ( α,α ) k  cos j α +1 k      = 0 , and (21) follo ws. Since the zeros t ( α,β ) k ,k tend to 1 as k tends to infinity , to prov e the theorem it suf fices to show that lim t → 1 m ( t ) exists. This foll ows from (21) and the follo wing two f acts w hich hold for all k ≥ 2 : (22) P ( α,α ) k ( t ( α +1 ,α +1) k − 1 ,k − 1 ) ≤ m ( t ) for all t ≥ t ( α +1 ,α +1) k − 1 ,k − 1 and (23) m ( t ) ≤ P ( α,α ) k +1 ( t ( α +1 ,α ) k ,k ) for all t ∈ [ t ( α +1 ,α +1) k − 1 ,k − 1 , t ( α +1 ,α +1) k ,k ] . Lowe r bounds for measurabl e chromatic numbers 13 Fact (22) follo ws from (18) and [1, (6.4.19)]. For establish ing fact (23) w e arg ue as follo ws: As in the proof of P ropos ition 7.1, we use (15) to show that P ( α +1 ,α ) k has exa ctly one zero in the interv al [ t ( α +1 ,α +1) k − 1 ,k − 1 , t ( α +1 ,α +1) k ,k ] , namely t ( α +1 ,α ) k ,k . From (16) we then see that t ( α +1 ,α ) k ,k is the only poin t in this interv al where P ( α,α ) k and P ( α,α ) k +1 coinci de. N o w it follo w s from the interla cing prope rty that P ( α,α ) k is increasing in the inte rv al and that P ( α,α ) k +1 is decre asing in the interv al, and we are done.  Cor ollary 8.2. W e have χ m ( R n ) ≥ 1 + ( j α +1 ) α 2 α Γ( α + 1) | J α ( j α +1 ) | , wher e α = ( n − 3) / 2 .  W e use this corollary to deri ve ne w lower bounds for n = 10 , . . . , 24 . W e giv e them in T abl e 8.1 . For n = 2 , . . . , 8 our bound s are worse than the e xistin g ones and for n = 9 our bound is 35 which is also the best known one. In fac t Oli vei ra and V allen tin [16] sho w , by differe nt methods, that the abov e bound is actually a bound for χ m ( R n − 1 ) . This t hen gi ves improv ed bou nds starting from n = 4 . W ith the use of addition al geometric argu m ents one can also get a ne w bound for n = 3 in this frame work. best lo wer boun d ne w lower bou nd n pre viously known for χ m ( R n ) for χ m ( R n ) 10 45 48 11 56 64 12 70 85 13 84 113 14 102 14 7 15 119 19 1 16 148 24 8 17 174 31 9 18 194 40 8 19 263 52 1 20 315 66 2 21 374 83 9 22 526 1060 23 754 1336 24 933 1679 T A B L E 8 . 1 . Lo wer bounds for χ m ( R n ) . W e can also use the corollary to show that our bound is exp onential in the di- mension . T o do so we use the inequaliti es (cf. [1, (4.14.1)] and [24, Section 15.3, p. 485 ]) j α +1 > j α > α 14 C. Bachoc, G. Nebe, F .M. de Oli veira Filho, F . V allenti n and (cf. [1, (4.9.13 )]) | J α ( x ) | ≤ 1 / √ 2 to obtai n ( j α +1 ) α 2 α Γ( α + 1) | J α ( j α +1 ) | > √ 2 α α 2 α Γ( α + 1) , and wit h Stirli ng’ s formula Γ( α + 1) ∼ α α e − α √ 2 π α we ha ve tha t the e xponenti al term is  e 2  α ∼ (1 . 16 5) n . 9. O T H E R S P AC E S In this se ction we want t o go b ack to our generali zation (3) of the thet a functi on and discuss its computa tion in more gener al situat ions than the one of the graph G ( n, t ) encounter ed in Section 6 . W e assume th at a compact gro up Γ acts conti nu- ously o n V . Then, the co m putatio n only depe nds on th e orthogon al decomposi tion of the space of L 2 -funct ions (24). 9.1. T wo -point homogeneous spaces. First, it is wor th notic ing that all results in Section 6 are va lid — one only has to use the approp riate zonal polynomials and approp riate vo lumes — for distance graphs in infinite, two- point homogen eous, compact metric spac es where edges are giv en by e xactly one distance. If one consi ders distance grap hs in infinite, compact, two-point homogeneo us metric spaces with s distanc es, then it is helpful to consid er a dual formulation of (9). It is an infinit e linear programming problem in dimensio n s + 1 which in the case of the unit sphe re has the followin g form: min  z 1 /ω n : z 1 + z t 1 + · · · + z t s ≥ ω 2 n , z 1 + z t 1 P ( α,α ) k ( t 1 ) + · · · + z t s P ( α,α ) k ( t s ) ≥ 0 for k = 1 , 2 , . . .  , where t 1 , . . . , t s are the inner produ cts defining the edges of our graph. 9.2. Symmetric spaces. N ext we may consider infinite compact metric spaces V which are not two-point homogeneou s bu t symmetric. Since the space L 2 ( V ) still has a m ultipli city-fr ee orthogona l decompo sition one gets a linear programming bound , but with the addi tional complicatio n that one has to work with multi va riate zonal p olyno mials. The most promine nt case of the Gra ssmann manifo ld was co n- sidere d by the first author in [2] in the conte xt of finding upper bound s for finite codes. 9.3. General homogeneous spaces. For the most general case one would ha ve multiplic ities m k in th e decompo sition of L 2 ( V ) which is gi ven b y the Peter-W e yl Theorem: (24) L 2 ( V ) = ( H 0 , 1 ⊥ . . . ⊥ H 0 ,m 0 ) ⊥ ( H 1 , 1 ⊥ . . . ⊥ H 1 ,m 1 ) ⊥ . . . , where H k ,l are Γ -irre ducibl e subspace s w hich are equ i valent whene ver thei r first inde x co incide s. In this case one uses Bochner ’ s character izatio n of the continuou s, Lowe r bounds for measurabl e chromatic numbers 15 Γ -in v ariant, positiv e kernel s giv en in [3, Section III] which yields a true semidefi- nite progra m ming problem for the computation of ϑ . 10. S E C O N D G E N E R A L I Z A T I O N In this sectio n we fi rst sho w how our generaliza tion relates to the theta func- tion of finite subgr aphs of G ( n, t ) . W e prov e that computing the theta function for any finite subg raph of G ( n, t ) does not giv e a better boun d than the one of Corol- lary 6.1. For thi s w e introduce a secon d genera lization of the theta function. Then we show ho w our second gene ralizat ion relates to the linear programming boun d of Delsarte . 10.1. F inite subgra phs. T o compute a bound for the measur able chromatic num- ber of the graph G ( n, t ) w e compute ϑ ( G ( n, t )) , which is an upper bound for α ( G ( n, t )) , and then ω n /ϑ ( G ( n, t )) is a lo wer bound for χ m ( G ( n, t )) . When G = ( V , E ) is a finite graph, this approach corresp onds to comput- ing ϑ ( G ) and using | V | /ϑ ( G ) as a lower bound for χ ( G ) . Ho w e ver , this is in genera l not the bes t bou nd we can obtain for χ ( G ) from the theta fu nction . Indeed, for a finite graph G , the so-call ed sandwich theor em says that α ( G ) ≤ ϑ ( G ) ≤ χ ( G ) (Theorem 3.1 only giv es the first inequ ality , Lov ´ asz [13, Proof of Corolla ry 3] gi ves the second ), where G is the complement of G , the graph with the same verte x set as G and in which two ve rtices are adjacent if and only if they are nonadja cent in G . Moreo ver , for a finite grap h G = ( V , E ) , w e ha ve (25) ϑ ( G ) ϑ ( G ) ≥ | V | (cf. Lov ´ asz [13, Coroll ary 2]). For some graphs (e.g ., stars), this inequality is stric t, hence in these cas es ϑ ( G ) wou ld provide us with a better lo wer bound for χ ( G ) than | V | /ϑ ( G ) would. But when V is homogene ous we actually hav e equality in (25) (cf. Lov ´ asz [13, Theorem 8]). In this case, both bounds for χ ( G ) coincid e. Something similar h appens for our infinit e distance grap h G ( n, t ) . The comple - ment of G ( n, t ) is the graph in which any two distinct points on the unit sphere whose inner product is not t are adjacent . W e cannot use our generalizati on of the theta function to d efine ϑ ( G ( n, t )) . Howe ver , we may use a dif ferent (and for finite graphs , equiv alent) definition of ϑ (cf. L ov ´ asz [13, T heorem 3]), which for a finite graph G = ( V , E ) is ϑ ( G ) = min  λ : K ∈ R V × V is posit i ve semidefini te , K ( x, x ) = λ − 1 for all x ∈ V , K ( x, y ) = − 1 if { x, y } ∈ E  . (26) The genera lization of this definition , applied to G ( n, t ) and w ith the symmetry tak en into accoun t, is describ ed belo w . W e choose to write ϑ ( G ( n, t )) instead 16 C. Bachoc, G. Nebe, F .M. de Oli veira Filho, F . V allenti n of ϑ ( G ( n, t )) to emphas ize that the two ways to define the theta func tion are not equi val ent for our infinite graph. So we hav e ϑ ( G ( n, t )) = min  λ : K ∈ C ( S n − 1 × S n − 1 ) is posit i ve , K is in v ariant unde r O ( R n ) , K ( x, x ) = λ − 1 for all x ∈ S n − 1 , K ( x, y ) = − 1 if x · y = t  . (27) By decomposing the kernel K with the help of the Jacobi polyn omials as done in Section 6, we may compu te the optimal valu e of the optimiz ation prob lem (27), and in doin g so we find out that ϑ ( G ( n, t )) ϑ ( G ( n, t )) = ω n , so that we ha ve the analo gue of ϑ ( G ) ϑ ( G ) = | V | for our infinite distan ce graph on the unit sphe re. This also provide s us with the connectio n to the theta function of finite sub- graphs of G ( n, t ) claimed in Section 6. If H = ( V , E ) is a finite subgr aph of G ( n, t ) , then ϑ ( H ) provide s a lower bound for χ ( H ) , which in turn is a lo wer bound for χ m ( G ( n, t )) . It could be that for some finite subgrap h H of G ( n, t ) this lo wer bound wou ld be better than the one provided by ϑ ( G ( n, t )) . This is, how- e ver , not the case. Ind eed, if K is an optimal solution for (27), the re stricti on of K to V × V is a feas ible solution to the optimiz ation prob lem (26 ) defining ϑ ( H ) , hence ϑ ( H ) ≤ ϑ ( G ( n, t )) , which is our bound for χ m ( G ( n, t )) . 10.2. D elsart e’ s linear progra mming boun d. The second generalizatio n ϑ of the theta function is clos ely related to the linear programming bound for finite codes establ ished by Delsa rte in [5] a nd put into th e frame work of group represe ntations, which we use here, by Kabatiansk y and L e ven shtein in [10]. Here we devi se an exp licit connection between these two bound s. T he connecti on between the linear progra mming boun d and the theta function was already obse rved by McEliece, Rodemich, Rumsey Jr . in [15] and independen tly by Schrijve r in [19] in the case of finite graph s. Consider the graph on the unit sphere where two distinct points are adjacen t whene ver thei r inner pr oduct lies i n the o pen interv al [ − 1 , t ] . W e denot e this gr aph by G ( n, [ − 1 , t ]) . Stable sets in the complemen t of this grap h are fi nite and con sist of poin ts on the unit sphere w ith minimal angu lar distance arccos t . No w the secon d generalizat ion (26) applied to G ( n, [ − 1 , t ]) is ϑ ( G ( n, [ − 1 , t ])) = inf  λ : K ∈ C ( S n − 1 × S n − 1 ) is posit i ve , K is in v ariant under O( R n ) , K ( x, x ) = λ − 1 for all x ∈ S n − 1 , K ( x, y ) = − 1 if x · y ∈ [ − 1 , t ]  . (28) W e safely w rite inf instead of min here becaus e we do not kno w if the infimum is attaine d. Lowe r bounds for measurabl e chromatic numbers 17 Pro p osition 10.1. Let C ⊆ S n − 1 be a subset of the unit spher e such that e very pair of distinct points in C has inner pr oduct lying in [ − 1 , t ] . Then its car dinality is at most ϑ ( G ( n, [ − 1 , t ])) . Pr oof. Let K be a kernel satisfying the conditions in (28 ). Then, by the positi vity of the conti nuous kernel K it follo ws that 0 ≤ X ( c,c ′ ) ∈ C 2 K ( c, c ′ ) = X c K ( c, c ) + X c 6 = c ′ K ( c, c ′ ) ≤ | C | K ( c, c ) − | C | ( | C | − 1) , so that | C | − 1 ≤ K ( c, c ) and we are done.  W e finish by sho wing how the original formu lation of the linear progra m ming bound can be obtained from (28). Using S choenb erg’ s characte rization (7) the semidefinit e programming problem (28) simplifies to the linear programming prob- lem inf  λ : f 0 ≥ 0 , f 1 ≥ 0 , . . ., ∞ X k =0 f k P ( α,α ) k (1) = λ − 1 , ∞ X k =0 f k P ( α,α ) k ( u ) = − 1 for all u ∈ [ − 1 , t ]  . W e can streng then it by requirin g P ∞ k =0 f k P ( α,α ) k ( u ) ≤ − 1 for all u ∈ [ − 1 , t ] . By restric ting f 0 = 0 the infimum is not ef fected. Then, after simplification, we get the line ar programming bound (cf. [6], [10]). inf { 1 + ∞ X k =1 f k : f 1 ≥ 0 , f 2 ≥ 0 , . . . , ∞ X k =1 f k P ( α,α ) k ( u ) ≤ − 1 for all u ∈ [ − 1 , t ]  . By Proposition 10.1 it giv es an uppe r bound for the maximal number of points on the unit sphere with minimal angul ar distance arccos t . A C K N O W L E D G E M E N T S W e thank Dion Gijswijt, Gil Kalai, T om Koor nwinder , Pablo Par rilo, and Lex Schrijv er for their help ful comments. R EF E R E N C E S [1] G.E. Andre ws, R. Aske y , R. Roy , Special functions , Cambridge Univ ersity Pr ess, 1999 . [2] C. Bachoc, Linear pr ogr amming bounds for codes in Gr assmannian spaces , IEEE T rans. Inf. Th. 52 (2006), 2111–212 5. [3] S. Bochner , Hi lbert distances and positive definite functions , Ann. of Math. 42 (1941), 647– 656. [4] N.G. de Bruijn, P . Er d ˝ os, A colour pr oblem for infinite graph s and a pr oblem in the theory of r elations , Indagationes Math. 13 (1951 ), 369–373. 18 C. Bachoc, G. Nebe, F .M. de Oli veira Filho, F . V allenti n [5] P . Delsarte, An algebraic appr oach to t he association schemes of coding theory , Philips R es. Rep. Suppl. (1973), vi+97. [6] P . 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N E B E , L E H R S T U H L D F ¨ U R M AT H E M A T I K , RW T H A A C H E N U N I V E R S I T Y , T E M P L E R - G R A B E N 6 4 , 5 2 0 6 2 A AC H E N , G E R M A N Y E-mail addr ess : nebe@math.rwth-a achen.de F. M . D E O L I V E I R A F I L H O , C E N T R U M V O O R W I S K U N D E E N I N F O R M A T I C A ( C W I ) , K RU I S - L A A N 4 1 3 , 1 0 9 8 S J A M S T E R DA M , T H E N E T H E R L A N D S E-mail addr ess : f.m.de.oliveira. filho@cwi.nl F. V A L L E N T I N , D E L F T I N S T I T U T E O F A P P L I E D M A T H E M AT I C S , T E C H N I C A L U N I V E R S I T Y O F D E L F T , P . O . B OX 5 0 3 1 , 2 6 0 0 G A D E L F T , T H E N E T H E R L A N D S E-mail addr ess : f.vallentin@tude lft.nl