Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

Asso ciation sc hemes related to univ ersally optima l configurations , Kerdo c k co des and extremal Euclidean line-sets Kanat Ab dukhalik o v ∗ Institute of Mathematics, Pushkin Str 125, Almat y 05001 0, Kazakhstan ab dukhalik ov @math.kz Eiic hi Bannai Graduate Sc ho ol of Mathematics, Kyush u Univ ersit y , Hak ozaki 6-10- 1, Higashi-ku, F ukuok a 8 12-8581, Japan bannai@math.kyush u-u.ac.jp Sho Suda Graduate Sc ho ol of Mathematics, Kyush u Univ ersit y , Hak ozaki 6-10- 1, Higashi-ku, F ukuok a 8 12-8581, Japan Abstract H. Cohn et. al. prop o s ed an asso ciation sc heme of 64 p oints in R 14 which is conjectured to be a univ ers ally optimal co de. W e show that this scheme has a genera liz ation in terms of Ke rdo ck co des, as well a s in ter ms of maximal rea l mutually unbiased bases. These schemes also re lated to extremal line-sets in E uclidean spaces and Barnes-W all lattices. D. de Caen and E. R. v an Dam co ns tructed tw o infinite series of formally dual 3 -class asso c iation schemes. W e explain this forma l dualit y by constructing t wo dual ab elian s chemes related to quaternar y linear Ker do ck and Pre parata co des. Keyw ords: u niv ersally optimal confi gurations, asso ciation sc hemes, dual sc hemes, Kerdo c k co des, Preparata co des, mutually un biased bases, Barnes-W all lattices. 1 In tro duc tion In [8] Henry Cohn and others defined and studied unive rs ally optimal configur ations in Euclidean spaces, spherical p oin t configurations that minimize broad class of fun ctions lik e p otent ial energy functions. T hey also obtained exp erimenta l results and conjectured that a some three class as- so ciation scheme on 64 p oints determines u niv ersally optimal configur ation in R 14 . This s cheme is uniquely determined by their parameters [2] and has automorphism group 4 3 : (2 × L 3 (2)), where 2 × L 3 (2) is the stabilize r of a p oint. It h as the follo wing fi rst and second eige n m atrices: P = Q =     1 14 42 7 1 − 6 6 − 1 1 2 − 2 − 1 1 − 2 − 6 7     . The s cheme generates a configuration of 64 v ectors on a spher e of squared radius 7 in R 14 . These vect ors generate inte gral lattice with automorph ism group 2 14 : (2 3 : L 3 (2)). Th e lattice ∗ Supp orted by Japan So ciety for the Promotion of Science 1 can b e obtained b y co nstr uction A from b inary shortened pro jectiv e [14,4,7 ] co de. Theta series of the lattice is equal to θ ( q ) = 1 + 28 q 4 + 1024 q 7 + 2156 q 8 + · · · . So the 64 vec tors are not minimal v ectors of the lattice. W e sho w that th e sc heme has a generalization in terms of binary and quaternary Kerdo c k an d Preparata co des, as w ell as in terms of maximal real m utually unbiased bases (MUB). S tarting from dou b ly shortened binary (sh ortened quaternary) Kerdo c k cod e of length N (resp. of length N/ 2), w here N = 2 m +1 with o dd m , one can construct a 3-class asso ciation scheme of size N 2 / 4, whic h leads to a sp h erical co de in R N − 2 of size N 2 / 4. As binary Kerdo c k co d e we understand a binary co de of length N obtained fr om a K er d o c k set [18], so Kerdo ck co de is inside of a second order Reed-Muller co d e (see for details S ection 3), and it has nonzero distances 2 m ± 2 ( m − 1) / 2 , 2 m , 2 m +1 . F urthermore, we define Ker d o c k-lik e code as a b inary co de of length N with nonzero distances ( N ± √ N ) / 2, N/ 2 and N (in particular, Kerdo c k cod es a re Kerdo c k-lik e co des). It is an op en question wh ether Kerd o c k-lik e co de is actually a Kerd o c k co de. W e do not eve n know whether N m ust ha ve the form 2 m +1 . An y collection of maximal mutually unbiased bases in R N is in one-to-one corresp ondence with Kerdo ck-lik e co d es. In particular, any Kerdo c k co de determines maximal MUB. W e sh o w that in fact any maximal MUB in R N (equiv ale ntly , any Kerdo ck-lik e cod e) determines a 3-class asso ciation s cheme in R N − 2 of size N 2 / 4 with th e same parameters as sc hemes obtained from Kerdo c k co d es (Theorem 5). D. de C aen and E. R. v an Dam [6] constructed tw o infinite ser ies of formally d ual 3-class asso ciation sc hemes, related to Kerd o c k sets. W e explain this formal dualit y by constr u cting t w o dual ab elian schemes related to quaternary linear Kerdo c k and P reparata co des (Theorem 1). The situatio n is similar to one, when the f ormal dualit y b et w een b inary n onlinear Kerdo c k and Preparata co des w as explained b y d ualit y b et we en quaternary lin ear Kerdo c k and Pr ep arata co des [17]. W e also n ote that maximal r eal m utu ally unbiased bases determine a 4-class asso ciation sc heme of size N 2 + 2 N in R N , as well as a 3-class asso ciation scheme of size N 2 / 2 in R N − 1 (whic h corresp onds to the association sc heme obtained from a shortened Kerdo c k-like code). R. L. Griess Jr. sho wed [15] that the 64 p oint co de and other tricosine co des can also b e constructed us ing minimal vec tors of Barnes-W all lattice. W e giv e later explanation of this phenomenon in terms of extremal line-sets and Kerdo c k co des. It seems that the schemes on shortened Kerdo ck co d es also might b e candidates for b eing unive rsally optimal (or optimal) configurations in R 2 m +1 − 2 . W e note that in R 14 for θ -co d e, θ = 1 7 , Lev enshtei n’s b ound is 69.6 (corresp onding scheme has 64 p oin ts). Similarly , in R 62 for θ -co de, θ = 3 31 , Lev ensh tein’s b ound is 108 1 (corresp ondin g sc h eme h as 102 4 p oints). The authors wish to thank H. Cohn , R. L. Griess Jr, O. Mu sin, A. Barg and C. Carlet for useful consultations. 2 Quaternary K erdo c k a nd Preparata co des and dual ab elian asso ciation sc hemes In this section w e construct tw o dual ab elian asso ciation sc hemes in terms of qu aternary linear Kerdo c k an d Preparata co des. By quaternary linear Kerdo ck and Preparata co d es w e mean the class of co d es determined in [7]. Recall that Z 4 -linear Kerdo ck co d e K and Z 4 -linear Pr eparata co de P are linear cod es o v er Z 4 of length q = 2 m = N / 2, m ≥ 3, m o dd: 0 ⊂ K ⊆ P ⊂ Z q 4 . 2 They are dual co d es: K ⊥ = P . Moreo v er, K = P for m = 3 and K 6 = P for m > 3. The image under the Gray map of th e qu aternary Ker d o c k (resp. Preparata) co de is b inary nonlinear Kerdo c k (resp. ”Preparata ”) co de. F or m = 3 the Gra y image of K = P is the famous binary nonlinear Nords trom-Robinson co de of length 16. Cons id er sh ortened Kerdo ck and pu nctured Preparata co des: 0 ⊂ K short ⊆ P punct ⊂ Z q − 1 4 = A. F or a co d e C , the punctured co de comes from d eleting the co ordin ate at p osition i , and the shortened co de fr om deleting the 0 at i from the w ords of th e sub co de of C ha ving 0’s at i . Since the automorphism group of Kerdo ck co d e acts transitive ly on co ord inates we can consider shortening and pun cturing at an y fixed (same) p osition. Note that K ⊥ short = P punct . Therefore, one has nondegenerate bilinear pairing ( K short , A/P punct ) → Z 4 , whic h gives us dualit y b et we en K short and A/P punct . W e ha ve | K short | = | A/P punct | = q 2 = 4 m . W e can consider A/P punct as c haracter group of K short or vice v ersa. Therefore, an ab elian asso ciation sc heme on K short defines dual ab elian sc heme on A/P punct (cosets of P punct ). Shortened Z 4 -Kerdo c k co de [7, 17] is a co de of length 2 m − 1, m o dd. It h as 4 m = N 2 / 4 co dew ords and nonzero cod ew ords ha ve Lee weig hts 2 m + 2 ( m − 1) / 2 , 2 m − 2 ( m − 1) / 2 , and 2 m . W e are goi n g to show that the foll owing r elations on the sh ortened Kerdo c k code will determine an ab elian 3 class asso ciation sc heme: ( x, y ) ∈        R 0 , if x − y h as w eigh t 0 , R 1 , if x − y h as w eigh t 2 m + 2 ( m − 1) / 2 , R 2 , if x − y h as w eigh t 2 m − 2 ( m − 1) / 2 , R 3 , if x − y h as w eigh t 2 m . (1) Cosets of punctured Z 4 -Preparata co de C = P punct ha ve Lee w eigh ts 0, 1 an d 2. F urthermore, for cosets a + C w e can c ho ose a = (0 , . . . , 0 , ± 1 , 0 , . . . , 0), a = (0 , . . . , 0 , +1 , . . . , − 1 , 0 , . . . , 0) or a = (0 , . . . , 0 , 2 , 0 , . . . , 0) (see Lemma 2). The follo wing relations ( x, y ) ∈        R ′ 0 , if x − y = C, R ′ 1 , if x − y = (0 , . . . , 0 , ± 1 , 0 , . . . , 0) + C , R ′ 2 , if x − y = (0 , . . . , 0 , +1 , . . . , − 1 , 0 , . . . , 0) + C, R ′ 3 , if x − y = (0 , . . . , 0 , 2 , 0 , . . . , 0) + C . (2) on A/C will d efine a three class association scheme whic h is dual to the previous sc heme. Theorem 1 The r elations (1) on c o dewo r ds of shortene d Z 4 -Ker do ck c o de define a thr e e class (ab elian) asso ciation scheme, with the first and the se c ond eigenmatric es given by: P =       1 ( N − 2 √ N )( N − 2) 8 ( N +2 √ N )( N − 2) 8 N 2 − 1 1 − √ N ( N − 4) 8 √ N ( N − 4) 8 − 1 1 √ N 2 − √ N 2 − 1 1 − N − 2 √ N 4 − N +2 √ N 4 N 2 − 1       , Q =      1 N − 2 ( N − 2)( N − 4) 4 N 2 − 1 1 − √ N − 2 √ N + 2 − 1 1 √ N − 2 − √ N + 2 − 1 1 − 2 − N 2 + 2 N 2 − 1      . 3 The r elations (2 ) on c osets of punctur e d Z 4 -Pr ep ar ata c o de define an asso ciation scheme which is dual to the former scheme, so the scheme has the fol lowing first and se c ond ei g enmatric es: P ′ = Q , Q ′ = P . If we tak e the particular class of K erdo c k and Pr eparata co des considered in [17] then th e sc heme has automorphism group 4 m : Aut ( K short ), where Aut ( K short ) ∼ = 2 × L 3 (2) for m = 3 and Aut ( K short ) ∼ = 2 × ( F ∗ 2 m : Aut ( F 2 m )) for m > 3. W e also note that the ab elian group K short is isomorph ic to the Galois ring GR (4 , m ) = Z 4 [ ξ ], ξ q − 1 = 1, q = 2 m . Isomorphism is giv en by map γ 7→ ( T r ( γ ξ 0 ) , T r ( γ ξ 1 ) , . . . , T r ( γ ξ q − 2 )), where γ ∈ GR (4 , m ) (see for details [17]). A t fi rst w e stu dy the structure of cosets of punctured Z 4 -Preparata co de C = P punct . Lemma 2 a) Ther e exists a p artition A/C = V 0 ∪ V 1 ∪ V 2 ∪ V 3 into four sets, wher e V 0 , V 1 , V 2 and V 3 ar e c osets of the form C , (0 , . . . , 0 , ± 1 , 0 , . . . , 0) + C , (0 , . . . , 0 , + 1 , 0 , . . . , 0 , − 1 , 0 , . . . , 0) + C and (0 , . . . , 2 , . . . , 0) + C , r esp e ctively. b) The pr evious statement r emains true i f the set V 2 is enumer ate d by elements of the form (0 , . . . , 0 , 1 , 0 , . . . , 0 , 1 , 0 , . . . , 0) + C and (0 , . . . , 0 , − 1 , 0 , . . . , 0 , − 1 , 0 , . . . , 0) + C . c) Final ly, V 2 c an b e enumer ate d by elements of the form (0 , . . . , 0 , 2 , 0 , . . . , 0 , 1 , 0 , . . . , 0) + C . Pr o of . a) First we note that | V 0 | = 1, | V 1 | = 2( q − 1), | V 2 | = ( q − 1)( q − 2), | V 3 | = q − 1, so | V 0 | + | V 1 | + | V 2 | + | V 3 | = q 2 = | A/ C | . It r emains for us to pro ve that element s of all V i are differen t. Minim um Lee we ight of Z 4 -Preparata co d e P is 6, therefore minimum Lee w eight of punctured co de C is either 4 (if there is an elemen t (2 , ± 1 , ± 1 , ± 1 , ± 1 , 0 , . . . , 0) ∈ P ) or 5. If the elemen ts of V i are not different, then C con tains elemen ts of the f orm (1 , 1 , − 1 , − 1 , 0 , . . . , 0) or (2 , 1 , − 1 , 0 , . . . , 0), w hic h means (2 , 1 , 1 , − 1 , − 1 , 0 , . . . , 0) ∈ P or (2 , 2 , 1 , − 1 , 0 , . . . , 0) ∈ P . These cases are not p ossible, since co dewords of P are zero sum vec tors ((1 , . . . , 1) ∈ K by definition and P = K ⊥ ) and the cod eword (0 , 0 , 2 , 2 , 0 , . . . , 0) of w eigh t 4 d o es not b elong to P . P arts b) and c) can b e prov ed similarly .  Pr o of of The or e m 1 . First we pro ve that the relations (2 ) on cosets of p unctured Z 4 -Preparata co de define an asso ciation sc heme. Consid er the follo wing elemen ts in C [ A/C ] (the group ring of the group A/C ov er the fi eld of co mp lex n umb ers): D i = X v ∈ V i Z v , 0 ≤ i ≤ 3 . W e will sh ow that the su balgebra < D 0 , D 1 , D 2 , D 3 > , generated by element s D 0 , D 1 , D 2 and D 3 , is Sc hurian. The follo w ing equalities are obtained ju st from c ounting elemen ts in D i : D 1 · ( D 0 + D 1 + D 2 + D 3 ) = 2( q − 1)( D 0 + D 1 + D 2 + D 3 ) , D 2 · ( D 0 + D 1 + D 2 + D 3 ) = ( q − 1)( q − 2)( D 0 + D 1 + D 2 + D 3 ) , D 3 · ( D 0 + D 1 + D 2 + D 3 ) = ( q − 1)( D 0 + D 1 + D 2 + D 3 ) , F urther, we ha ve ( D 0 + D 3 ) 2 = ( D 0 + D 3 )( D 0 + D 3 ) = q ( D 0 + D 3 ) , whic h implies D 2 3 = ( q − 1) D 0 + ( q − 2) D 3 . 4 Lemma 2 b) implies that D 2 1 = 2( q − 1) D 0 + 4 D 2 + 2 D 3 . Finally , by lemma 2 c) we ha ve D 3 D 1 = D 1 + 2 D 2 . Matrices of m ultiplications by D 1 , D 2 , D 3 with r esp ect to basis D 0 , D 1 , D 2 , D 3 are given b y ρ 1 =     0 2 q − 2 0 0 1 0 2( q − 2) 1 0 4 2( q − 4) 2 0 2 2( q − 2) 0     , ρ 2 =     0 0 ( q − 1)( q − 2) 0 0 2( q − 2) ( q − 4)( q − 2) q − 2 1 2( q − 4) q 2 − 6 q + 12 q − 3 0 2( q − 2) ( q − 3)( q − 2) 0     , ρ 3 =     0 0 0 q − 1 0 1 q − 2 0 0 2 q − 3 0 1 0 0 q − 2     . It is easy to s ee that vec tors v 1 = (1 , 2 q − 2 , ( q − 1)( q − 2) , q − 1), v 2 = (1 , − √ 2 q − 2 , √ 2 q +2 , − 1), v 3 = (1 , √ 2 q − 2 , − √ 2 q + 2 , − 1), v 4 = (1 , − 2 , − q + 2 , q − 1) are common left eigen ve ctors for matrices ρ i = t B i . They are strok es of the matrix P ′ = Q =     1 2 q − 2 ( q − 1)( q − 2) q − 1 1 − √ 2 q − 2 √ 2 q + 2 − 1 1 √ 2 q − 2 − √ 2 q + 2 − 1 1 − 2 − q + 2 q − 1     . No w w e are going to prov e that relations (1) defin e an asso ciation sc heme dual to sc heme (2). The latter ab elian scheme determines a dual sc heme on K short , with p artition K short = V ′ 0 ∪ V ′ 1 ∪ V ′ 2 ∪ V ′ 3 . W e pro v e that this partition corresp ond s to sets of co dewords of Lee w eights 0, 2 m + 2 ( m − 1) / 2 , 2 m − 2 ( m − 1) / 2 , and 2 m resp ectiv ely . Indeed, according to [11, section 4.7.1], for an y elemen t u ∈ V ′ j w e should ha ve Q j k = X v ∈ V k i − ( v,u ) , where i = √ − 1. Such equ ations d etermine elemen ts in V ′ j uniquely . F or example, let u s tak e co dew ord u ∈ K short of Lee we ight 2 m + 2 ( m − 1) / 2 (other cases can b e consid er ed analogo u sly). Then co dew ord u has form (2 a , 1 b , ( − 1) c , 0 q − 1 − a − b − c ), 2 a + b + c = 2 m + 2 ( m − 1) / 2 . S ince 2 u = (0 a , 2 b , 2 c , 0 q − 1 − a − b − c ) ∈ K short , w e ha ve b + c = 2 m − 1 and a = ( q + √ 2 q ) / 4. F or k = 1 w e ha ve V 1 = { (0 , . . . , 0 , ± 1 , 0 , . . . , 0) + C } , | V 1 | = 2( q − 1), and X v ∈ V 1 i − ( v,u ) = q + √ 2 q 4 i 2 + bi − 1 + ci + ( q − √ 2 q 4 − 1) + q + √ 2 q 4 i 2 + bi + ci − 1 + ( q − √ 2 q 4 − 1) = − p 2 q − 2 . 5 Therefore, u ∈ V ′ 1 . Finally w e show that one can use a shortening (puncturing) at an y p osition. It is e n ough to sho w that the automorphism group of Kerdo c k code is transitiv e on co ordin ates. It ju st f ollo ws from the defin ition of Z 4 -Kerdo c k cod e [7 ]. Let V b e a m -dimensional v ector space o ver F 2 and R b e a binary symmetric m × m matrix. The map T R : V → Z 4 is giv en b y T R ( v ) = m X i =1 R ii b v 2 i + 2 X i 0) are mutually un biased. One ca n conv ert L to a binary co de C b y changing 1 / √ N and − 1 / √ N to 0 and 1, resp ectiv ely . Then it is easy to see that C is a Kerd o c k-lik e co de (the conditions ( x, y ) = ± 1 / √ N , 0, 1 for x , y ∈ L mean exactly that distances d ( x ′ , y ′ ) = ( N ∓ √ N ) / 2, N/ 2, N for corresp onding images x ′ , y ′ ∈ C ). In the co nstr uction of a sso ciatio n sc hemes w e used doubly shortened bin ary Kerdock co des. No w we do similar pro cedure for m utually unbiase d bases: ta ke v ectors fr om L with 1 / √ N in t w o fixed co ordinates and then delete these co ordin ates. W e will get a configuration of N 2 / 4 v ectors in R N − 2 of equal length, su c h that cosines of an gles b et w een distinct vect ors are equal to − √ N − 2 N − 2 , − 2 N − 2 and √ N − 2 N − 2 . 8 Theorem 5 L et M b e a maximal mutual ly unbiase d b ases in R N and X = M ∪ ( − M ) . F or u, v ∈ X such that ( u, v ) = 0 , we put Y ′ := { x ∈ X | ( u, x ) = ( v , x ) = 1 √ N } , and let Y b e the ortho gonal pr oje ction of ve ctors Y ′ to h u, v i ⊥ r esc ale d to make a spheric al c o de. Then Y is an asso c i ation scheme with the first and se c ond eigenmatric es given by: P =       1 ( N − 2 √ N )( N − 2) 8 ( N +2 √ N )( N − 2) 8 N 2 − 1 1 − √ N ( N − 4) 8 √ N ( N − 4) 8 − 1 1 √ N 2 − √ N 2 − 1 1 − N − 2 √ N 4 − N +2 √ N 4 N 2 − 1       , Q =      1 N − 2 ( N − 2)( N − 4) 4 N 2 − 1 1 − √ N − 2 √ N + 2 − 1 1 √ N − 2 − √ N + 2 − 1 1 − 2 − N 2 + 2 N 2 − 1      . Pr o of . W e p u t α = − √ N − 2 N − 2 , β = √ N − 2 N − 2 , γ = − 2 N − 2 . S ince A ( Y ) = { α, β , γ } , Y is a 3- distance s et. The annih ilator p olynomial F ( x ) := Q α ∈ A ( Y ) x − α 1 − α has the Geg enbauer p olynomial expansion F ( x ) = 4 N 2 Q 0 ( x ) + 2( N 2 + 6)( N − 2) N 3 ( N − 1) Q 1 ( x ) + ( N − 2) 3 ( N + 3) N 3 ( N − 1) Q 2 ( x ) + 6( N − 2)( N − 3) N 2 ( N − 1) Q 3 ( x ) . As | Y | = N 2 / 4 , Theorem 6.5 of [14] imp lies that Y is a spherical 3-design. By Lemma 7.3 of [14], for 0 ≤ i, j ≤ 2, i + j 6 = 4 and z := ( ξ , η ), the in tersection n umb ers p α,β ( ξ , η ) s atisfy th e linear equation X x,y ∈ A ( Y ) x i y j p x,y ( ξ , η ) = N 2 4 F i,j ( z ) − z i − z j + δ 1 ,z , where t i = i X k =0 f i,k Q k ( t ) and F i,j ( t ) = min ( i,j ) X k =0 f i,k f j,k Q k ( t ). No w let z = ( ξ , η ) b e fixed. Then w e get f ollo win g equation:             1 1 1 1 1 1 1 1 α β γ α β γ α β α 2 β 2 γ 2 α 2 β 2 γ 2 α 2 β 2 α α α β β β γ γ α 2 αβ αγ αβ β 2 β γ αγ β γ α 3 αβ 2 αγ 2 α 2 β β 3 β γ 2 α 2 γ β 2 γ α 2 α 2 α 2 β 2 β 2 β 2 γ 2 γ 2 α 3 α 2 β α 2 γ αβ 2 β 3 β 2 γ αγ 2 β γ 2                         p α,α ( ξ , η ) p β ,α ( ξ , η ) p γ ,α ( ξ , η ) p α,β ( ξ , η ) p β ,β ( ξ , η ) p γ ,β ( ξ , η ) p α,γ ( ξ , η ) p β ,γ ( ξ , η )             =             F 0 , 0 ( z ) − z 0 − z 0 + δ 1 ,z − γ 0 p γ ,γ ( ξ , η ) F 1 , 0 ( z ) − z 1 − z 0 + δ 1 ,z − γ 1 p γ ,γ ( ξ , η ) F 2 , 0 ( z ) − z 2 − z 0 + δ 1 ,z − γ 2 p γ ,γ ( ξ , η ) F 0 , 1 ( z ) − z 0 − z 1 + δ 1 ,z − γ 1 p γ ,γ ( ξ , η ) F 1 , 1 ( z ) − z 1 − z 1 + δ 1 ,z − γ 2 p γ ,γ ( ξ , η ) F 2 , 1 ( z ) − z 2 − z 1 + δ 1 ,z − γ 3 p γ ,γ ( ξ , η ) F 0 , 2 ( z ) − z 0 − z 2 + δ 1 ,z − γ 2 p γ ,γ ( ξ , η ) F 1 , 2 ( z ) − z 1 − z 2 + δ 1 ,z − γ 3 p γ ,γ ( ξ , η )             , 9 where det             1 1 1 1 1 1 1 1 α β γ α β γ α β α 2 β 2 γ 2 α 2 β 2 γ 2 α 2 β 2 α α α β β β γ γ α 2 αβ αγ αβ β 2 β γ αγ β γ α 3 αβ 2 αγ 2 α 2 β β 3 β γ 2 α 2 γ β 2 γ α 2 α 2 α 2 β 2 β 2 β 2 γ 2 γ 2 α 3 α 2 β α 2 γ αβ 2 β 3 β 2 γ αγ 2 β γ 2             = ( α − β ) 6 ( α − γ ) 4 ( β − γ ) 4 6 = 0 . In case of ( ξ , η ) = 1 w e hav e p γ ,γ ( ξ , η ) = N / 2 − 1. In case of ( ξ , η ) = α or β , we hav e p γ ,γ ( ξ , η ) = 0. Finally , in case of ( ξ , η ) = γ we ha ve p γ ,γ ( ξ , η ) = N / 2 − 2. Therefore, in tersection num b ers p x,y ( ξ , η ) are determined uniquely b y p γ ,γ ( ξ , η ). Hence Y is an asso ciation sc heme. Th is completes the pro of of Theorem 5.  Remark. In Theorem 7.4 of [14] it is men tioned that if t ≥ 2 s − 3, then for an y fixed z = ˜ γ = ( ξ , η ), the in tersection num b ers p ˜ α, ˜ β ( ξ , η ) are uniquely determined b y p ˜ γ , ˜ γ ( ξ , η ). Our claim is that in tersection num b ers p ˜ α, ˜ β ( ξ , η ) are uniquely determined by p γ ,γ ( ξ , η ) with a suitable γ which is not necessarily equal to ˜ γ . So, our argumen t is sligh tly general than in Theorem 7.4 of [14]. Here we mention some more useful inform ation. W e can easily calculate the intersectio n matrices B i = ( p k i,j ) 0 ≤ j ≤ 3 , 0 ≤ k ≤ 3 and Krein parameter matrices B ∗ i = ( q k i,j ) 0 ≤ j ≤ 3 , 0 ≤ k ≤ 3 . Namely , they are giv en as follo ws: B 1 =      0 1 0 0 ( N − 2 √ N )( N − 2) 8 ( N +2 √ N )( N − 7 √ N +12) 16 ( N − 2 √ N )( N − √ N − 4) 16 ( N − 2 √ N )( N − 2 √ N − 4) 16 0 ( N +2 √ N )( N − √ N − 4) 16 ( N − 2 √ N )( N + √ N − 4) 16 N ( N − 4) 16 0 N − 2 √ N − 4 4 N − 2 √ N 4 0      , B 2 =      0 0 1 0 0 ( N +2 √ N )( N − √ N − 4) 16 ( N − 2 √ N )( N + √ N − 4) 16 N ( N − 4) 16 ( N +2 √ N )( N − 2) 8 ( N +2 √ N )( N + √ N − 4) 16 ( N − 2 √ N )( N +7 √ N +12) 16 ( N +2 √ N )( N +2 √ N − 4) 16 0 N +2 √ N 4 N +2 √ N − 4 4 0      , B 3 =      0 0 0 1 0 N − 2 √ N − 4 4 N − 2 √ N 4 0 0 N +2 √ N 4 N +2 √ N − 4 4 0 N 2 − 1 0 0 N 2 − 2      ; B ∗ 1 =     0 1 0 0 N − 2 0 4 2 0 N − 4 N − 8 N − 4 0 1 2 0     , B ∗ 2 =     0 0 1 0 0 N − 4 N − 8 N − 4 ( N − 2)( N − 4) 4 ( N − 4)( N − 8) 4 N 2 − 12 N +48 4 ( N − 4)( N − 6) 4 0 N 2 − 2 N 2 − 3 0     , B ∗ 3 =     0 0 0 1 0 1 2 0 0 N 2 − 2 N 2 − 3 0 N 2 − 1 0 0 N 2 − 2     . 10 The original asso ciation sc heme X of size N 2 + 2 N in R N (attac hed to a r eal MUB) is a class 4 asso ciation s c heme with the follo w in g parameters. Note that this is a 5-spherical design and of degree s = 4. It is in teresting to n ote that they are Q -p olynomial asso ciation sc hemes (and n ot P -p olynomial asso ciation sc heme f or N ≥ 4). The reader is referred to Ba nn ai-Bannai [3] for more d etails, where this fact was first noticed. These asso ciation sc hemes are p ossible candidates of universall y optimal co des in the sense of Cohn-Ku mar [9]. B 1 =        0 1 0 0 0 N 2 2 ( N + √ N )( N − 2) 4 N 2 4 ( N − √ N )( N − 2) 4 0 0 N − 1 0 N − 1 0 0 ( N − √ N )( N − 2) 4 N 2 4 ( N + √ N )( N − 2) 4 N 2 2 0 0 0 1 0        , B 2 =       0 0 1 0 0 0 N − 1 0 N − 1 0 2( N − 1) 0 2( N − 2) 0 2( N − 1) 0 N − 1 0 N − 1 0 0 0 1 0 0       , B 3 =        0 0 0 1 0 0 ( N − √ N )( N − 2) 4 N 2 4 ( N + √ N )( N − 2) 4 N 2 2 0 N − 1 0 N − 1 0 N 2 2 ( N + √ N )( N − 2) 4 N 2 4 ( N − √ N )( N − 2) 4 0 0 1 0 0 0        , B 4 =       0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0       ; B ∗ 1 =        0 1 0 0 0 N 0 2 N N +2 0 0 0 N − 1 0 N − 1 0 0 0 N 2 N +2 0 N 0 0 0 1 0        , B ∗ 2 =        0 0 1 0 0 0 N − 1 0 N − 1 0 ( N +2)( N − 1) 2 0 ( N +2)( N − 2) 2 0 ( N +2)( N − 1) 2 0 N ( N − 1) 2 0 N ( N − 1) 2 0 0 0 N 2 0 0        , B ∗ 3 =        0 0 0 1 0 0 0 N 2 N +2 0 N 0 N ( N − 1) 2 0 N ( N − 1) 2 0 N 2 2 0 N 3 2( N +2) 0 N ( N − 2) 2 0 N 2 0 N 2 − 1 0        , B ∗ 4 =       0 0 0 0 1 0 0 0 1 0 0 0 N 2 0 0 0 N 2 0 N 2 − 1 0 N 2 0 0 0 N 2 − 1       ; 11 P =        1 N 2 2 2( N − 1) N 2 2 1 1 N 3 2 2 0 − N 3 2 2 − 1 1 0 − 2 0 1 1 − √ N 0 √ N − 1 1 − N 2( N − 1) − N 1        , Q =        1 N ( N − 1)( N +2) 2 N 2 2 N 2 1 √ N 0 − √ N − 1 1 0 − N 2 − 1 0 N 2 1 − √ N 0 √ N − 1 1 − N ( N − 1)( N +2) 2 − N 2 2 N 2        . The intermediate asso ciation sc heme Z b et we en X and Y , and of size N 2 / 2 in R N − 1 , where Z is obtained from v ectors Z ′ = { x ∈ X | ( x, u ) = 1 / √ N } b y orthogonal pro jection to h u i ⊥ for an y fixed u ∈ X and rescaling to mak e a spherical cod e, is a class 3 association sc h eme. Th is is a spherical 3-design and of degree s = 3 with the foll owing parameters. I t is in teresting to note that th ey are Q -p olynomial asso ciation sc hemes (and not P -p olynomial asso ciation s c heme for N ≥ 4.) It seems that this example was already recognized in th e list of W. Martin’s home- page (of suc h asso ciation sc hemes) as th ose coming f rom the linked symmetric designs. Th ese asso ciation sc h emes are also p ossible candidates of universally optimal co d es in the sense of Cohn-Kumar [9]. Here w e describ e the parameters of the asso ciation scheme Z . B 1 =       0 1 0 0 ( N − √ N )( N − 2) 4 ( N − 3 √ N )( N − 4) 8 ( N − √ N )( N − 4) 8 ( N − 2 √ N )( N − 2) 8 0 ( N + √ N )( N − 4) 8 ( N − √ N )( N − 4) 8 N ( N − 2) 8 0 ( √ N − 2)( √ N +1) 2 N − √ N 2 0       , B 2 =       0 0 1 0 0 ( N + √ N )( N − 4) 8 ( N − √ N )( N − 4) 8 N ( N − 2) 8 ( N + √ N )( N − 2) 4 ( N + √ N )( N − 4) 8 ( N +3 √ N )( N − 4) 8 ( N +2 √ N )( N − 2) 8 0 N + √ N 2 ( √ N − 1)( √ N +2) 2 0       , B 3 =      0 0 0 1 0 ( √ N − 2)( √ N +1) 2 N − √ N 2 0 0 N + √ N 2 ( √ N − 1)( √ N +2) 2 0 N − 1 0 0 N − 2      ; B ∗ 1 =     0 1 0 0 N − 1 0 2 0 0 N − 2 N − 4 N − 1 0 0 1 0     , B ∗ 2 =     0 0 1 0 0 N − 2 N − 4 N − 1 ( N − 2)( N − 1) 2 ( N − 4)( N − 2) 2 N 2 − 6 N + 12 2 ( N − 4)( N − 1) 2 0 N 2 − 1 N 2 − 2 0     , 12 B ∗ 3 =     0 0 0 1 0 0 1 0 0 N 2 − 1 N 2 − 2 0 N 2 − 1 0 0 N 2 − 2     . The first and the second eigenmatrices are the same as in Prop osition 4. Remarks. (1) W e ha v e association schemes X , Z , Y of sizes N 2 +2 N , N 2 / 2, N 2 / 4 (resp ectiv ely) in R N , R N − 1 , R N − 2 (resp ectiv ely), wh ere N m ust b e an ev en p ow er of 2. Cu rrent ly all of them are p ossible candidates of universally optimal co d es in the sense of Cohn-Ku mar, at least f or N ≥ 16. It is s h o wn in Cohn et. al. [10 ] that X in R 4 (i.e., for N = 4) is n ot universally optimal. (It is an op en question whether it is optimal or not.) Z for R 3 (i.e., for N = 4) is not unive rsally optimal n or optimal. On the other hand, Y for R 2 (i.e., for N = 4) is unive rsally optimal. Although it is a wild guess without fi rm grou n d, w e think Y ma y b e most likely to b e unive rsally optimal among X , Z , and Y . (2) It is an in teresting op en question whether these asso ciatio n sc hemes are un iquely determined b y the parameters. The u n iqueness of Y for R 14 (i.e., f or N = 16 ) was obtained in [2 ]. On th e other hand, the un iqueness of X for R 16 (i.e., for N = 1 6) w as prov ed by Akio Nak am ura [22 ] in h is masters degree th esis of Kyush u Univ ersity in 1997 (it follo ws also from the un iqueness of the Nordstrom-Robin s on co de). W e note that th e u niqueness of Z for R 15 (i.e., f or N = 16 ) is also obta ined . The clai m is essen tially obtained in [21], it can b e pr ov ed also by metho d of [22]. So, it would b e in teresting wh at will happ en in particular for N = 64 for X , Z , and Y . Th e result of Kan tor [19] implies that if N = 2 m +1 with o dd m , and if m is not a prime, then th ere are non-isomorphic line systems, and so there are non-isomorph ic asso ciation sc hemes with the same parameters, i.e., the uniqueness is break do wn. (3) Quite recen tly it w as shown [5] that the pr ob lem of constructing of s pairwise mutually unbiase d bases in K n ( K = R or K = C ) is equiv alen t to the problem of constructing of s C artan subalgebras of s l n ( K ) that are p airwise orthogonal with resp ect to Killing form and are closed under the adjoin t op eration. In particular, a complete collecti on of m utu ally unbiased bases in C n is equiv alen t to a n orthogonal decomposition of Lie algebra sl n ( C ), close d un der the adjoin t op eration. So there is a link to th e well-dev elop ed theory [20] of orthogonal decomp ositions of Lie algebras. 5 Extremal line-sets and Barnes-W all lattices In this section w e discuss connections b et wee n m utually unbiase d bases, extremal line-sets in R N with prescrib ed angles and minim um ve ctors of Barnes-W all lattices. Fix any p ositiv e int eger N > 1. Let M b e a set of un it vecto rs in R N , s uc h that | ( a, b ) | ∈ { 0 , 1 / √ N } for all a 6 = b in M (so, in particular, M ∩ ( − M ) = ∅ ). Then | M | ≤ N ( N + 2) / 2, and if | M | reac h es this upp er b ound , then M is a set of N / 2 + 1 mutually un b iased bases [7, Prop osition 3.12]. Constru ctions of such extremal line-sets are kno wn only for N = 2 m +1 , m o dd [7]. Our final observ ation is that known constructions of extremal line-sets (or, equiv alen tly , maximal sets of mutually u n biased bases) are connected to the minimum v ectors of Barnes-W all lattice s. W e sho w that vecto rs of known maximal real MUB afte r suitable rescaling will b ecome minimal v ectors of a Barnes-W all lattice. Therefore, v ectors of asso ciation schemes X , Y , Z can b e obtained from a set of minimal v ectors of the Barnes-W all lattices. First we recall the construction fr om [7]. Lab el the standard basis of R N as e v , with v ∈ 13 V = Z m +1 2 . F or b ∈ V , defin e the p erm u tation matrix X ( b ) and diagonal matrix Y ( b ) as follo ws: X ( b ) : e v 7→ e v + b and Y ( b ) := d iag[( − 1) b · v ] . The group s X ( V ) := { X ( b ) | b ∈ V } and Y ( V ) := { Y ( b ) | b ∈ V } are con tained in O ( R N ) and are isomorphic to the additiv e group V . Let E := h X ( V ) , Y ( V ) i . Then E = 2 1+2( m +1) + is an extrasp ecial 2-group of order 2 1+2( m +1) and E = E / Z ( E ) is elemen tary ab elian group of order 2 2( m +1) . W e id en tify the cent er Z ( E ) of E with Z 2 and consid er the map Q : E → Z 2 defined b y Q ( e ) = e 2 for any e ∈ E and any preimage e of e in E . Then Q is a non-singular quad r atic form on E . So E is an Ω + (2 m + 2 , 2)-space. Th e action of E on R N can b e extended to the action of the group 2 1+2( m +1) + Ω + (2 m + 2 , 2). The sp ace E cont ains (2 m +1 − 1)(2 m + 1) sin gular p oin ts. An orthogonal s pread of E is a family Σ of 2 m + 1 totally singular ( m + 1)-spaces suc h that every singular p oin t of E b elongs to exactly one mem b er of Σ. Let A b e a sub group of E su c h that its image A in E is totally sin gu lar ( m + 1)-space of E . T hen th e set F ( A ) of A -irr educible su b spaces of R N is an orthogonal frame: a set of 2 m +1 pairwise orthogonal lines through the origin. F or an orthogonal spread Σ of the Ω + (2 m + 2 , 2)- space E w e let F (Σ) := [ A ∈ Σ F ( A ) . Then F (Σ) consists of 2 m +1 (2 m + 1) lines of R N suc h that, if u 1 and u 2 are u nit v ectors in different mem b ers of F (Σ), th en | ( u 1 , u 2 ) | = 0 or 2 − ( m +1) / 2 . Therefore, F ( A ) determines orthonormal basis and F (Σ) determines a set of N/ 2 + 1 m utually unbiased b ases. These line-se ts F (Σ) are extremal in the sense that |F (Σ) | meets an upp er b ound obtained in [13] for line-sets in R N with prescrib ed angles. The binary Kerdo ck co de K (Σ) can b e reco vered [7] from F (Σ): K (Σ) = { ( c v ) v ∈ Z N 2 | h (( − 1) c v ) v i ∈ F (Σ) } . T ak e un it v ectors from line-set F (Σ), rescale them to v ectors of n orm √ N , then these ve ctors will b e minimum vect ors of a Barnes-W all lat tice. In d eed, w e note that for o dd m the minim u m norm o f Barnes-W all lattic e is √ N , the au tomorp h ism group is G = 2 1+2( m +1) Ω + (2 m + 2 , 2), G acts transitiv ely [15] on the set of minim u m vec tors, the v ector c = N − 1 / 4 P v ∈ V e v is a minim um v ector, an d any ev en lattice of rank N inv ariant under the group G is similar to a Barnes-W all lattice [15]. So all th e minim um v ectors of Barnes-W all lattice are obtained from c b y act ion of the g rou p G . On the other hand, in notatio n s of [7] w e ha ve c = N 1 / 4 e ∗ b for b = 0, and any un it v ector of F (Σ) is obtained from e ∗ b b y action of some elemen t of G . References [1] B. Ballinger, G. Blekherman, H. Cohn, N. 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