R894 Phi/ips Res. Repts 30, 91*-105*, 1975Issue in honour of C. J. Bouwkamp

BOUNDS FOR SYSTEMS OF LINES,AND JACOBI POLYNOMIALS

by P. DELSARTE, J. M. GOETHALSMBLE Research Laboratory

Brussels, Belgium

and J. J. SEIDEL

Technological University EindhovenEindhoven, The Netherlands

(Received November 13, 1974)

Abstract

Bounds are obtained for the cardinality of sets of lines having a pre-scribed number of angles, both in real and in complex Euclidean n-space.Extremal sets provide combinatorial configurations with a particularalgebraic structure, such as association schemes and regular two-graphs.The bounds are derived by use of matrix techniques and the additionformula for Jacobi polynomials.

1. Introduction

We consider sets of lines in real and in complex Euclidean n-space having aprescribed number of angles. In the case of one angle, two types of bounds areknown for the cardinality of such sets: one in terms of the angle and thedimension (cf. refs 13 and 12), the other in terms of the dimension only (theGerzon-bound, cf. ref. 12). In the present paper both types of bounds aregeneralized. The special bound (cf. table I) uses the values of the admittedangles. The absolute bound (cf. table II) uses the number of such angles, nottheir values.The essential tool in obtaining these results is the addition formula for Jacobi

polynomials. The classical addition formula for Gegenbauer polynomials (cf.ref. 9), was recently generalized to Jacobi polynomials by Koornwinder 10.11).

In certain linear spaces of harmonic polynomials this formula is interpreted asa (Hermitean) inner product. Sets ofvectors on the unit sphere in R", and in en,are characterized in terms of the matrices of their inner products. Thus, prop-erties of such sets are related to properties of Jacobi polynomials, and thetechniques of refs 7 and 8 may be used.Of particular interest are the sets of lines whose cardinality equals a bound.

In the case of one angle these sets are regular two-graphs 15.16); sometimes,they provide a combinatorial setting for interesting simple groups. Also in thegeneral case the extremal sets give rise to combinatorial configurations withinteresting algebraic properties, such as association schemes 2.8).

92* P. DELSARTE. J. M. GOETHALS AND J. J. SEIDEL

The cases R" and en are treated separately in secs 2 and 3, and simultaneouslyfrom sec. 4 on. The two families of Jacobi polynomials

{Qo ••(x), Ql,.(X), Q2,.(X), ... },

for e = ° and for e = 1, are defined by recurrence relations in sec. 2. Thevalue Qk,.(I) equals the dimension of the space Harm of the harmonic poly-nomials in n variables of the corresponding degrees. In theorem 3.3 the addi-tion formula is stated in terms of the Jacobi polynomials and an orthogonalbasis of Harm.

Let A be a finite subset of the interval [0, 1[, and let X be any finite subsetof the unit sphere Qn having the property that I(~, 1])12 belongs to A for all~ =1= 1]EX. In sec. 4 the characteristic matrices Hk,. are defined from X andan orthonormal basis of Harm. The crucial theorem 4.4 yields an inequalityfor lXi in terms of the Jacobi polynomials, the characteristic matrices, and apolynomial F(x) which behaves suitably for any x E A. This theorem is ap-plied in sec. 5 to the annihilator polynomial of the set A, yielding the specialbounds of theorem 5.2 and table I. In sec. 6 the characteristic matrices Ho,.,Hi,., ... , Hs-.,., with lAl = s, are combined into the matrix H., and ap-plication of theorem 4.4 yields the absolute bounds of theorem 6.1 and table Il.Several examples are given, such as those related to the simple groups ofConway 4) and Rudvalis, cf. ref. 5.In the final section 7 the linear spaces Ao and Ai are defined. A sufficient

condition for these spaces to be algebras is given in theorem 7.4, which appliesif the special bound and if the absolute bound is achieved, in theorems 7.5and 7.6, respectively.

2. Jacobi polynomialsFor each of the cases R" and en, with n ~ 2, and for e E {O,I}, we define

the family {Qo,.(x), Ql,.(X), ... } of polynomials Qk .•(X) in one real variable x.These are Jacobi polynomials, and share certain properties. We take % = °and 00 = 1.

Definition 2.1. For e E {O,I} and integer k ~ 0, the polynomials Qk ••(X) aredefined by the recurrence relations

Ak+l Qk+1.0(X) = x Qk.l(X) - (1- ilk) Qk.O(X),/kk+l Qk+1.1(X) = Qk+1.0(X) - (1- /kk) Qk.l(X),with the initial values Q-l,.(X) = 0, Qo.o(x) = 1.

For the case R", the coefficients are given by

2k 2k+lilk= , #k=---

n + 4k-2 n + 4k

BOUNDS FOR SYSTEMS OF LINES, AND JACOBI POLYNOMIALS 93*

The first polynomials are

Qo,o(x) = 1,

(n + 2) (nx-l)Ql,O(X) = 2 '

n (n + 6) ((n + 2) (n + 4) x2 - 6 (n + 2) x + 3)Q2,O(X) = 24 '

QO,l(X) = n,

n (n + 4) ((n + 2) x - 3)Ql,l(X) = 6 '

n (n + 2) (n + 8) ((n + 4) (n + 6) X2 -10 (n + 4) x + 15)Q2,l(X) = 120 . '

It follows that Qo,o(l) = 1 and, for k + 8 ~ 1,

(n + 2k +8-1) (n + 2k + 8-3)Qk,.(I) = - ,

n-l n-l

For the case en, the coefficients are given by

k k+lAk = n + 2k _ l' Ilk = n + 2k .

The first polynomials are

Qo,o(X) = 1,

Ql,O(X) = (n + l)(nx-l),

n (n + 3) (n + 1) (n + 2) X2 - 4 (n + 1) x + 2)Q2,O(X) = 4 , '

QO,l(X) = n,

n en + 2)((n + 1) x-2)Ql,l(X) = 2 '

n (n + 1) (n + 4) ((n + 2) (n + 3) X2 - 6 (n'+ 2) x + 6)Q2,l(X) = 12 .

94* P. DELSARTE,. J. M. GOETHALS AND J. J. SEIDEL

It follows that

Qk,.(I) = (n + k + e - I) (n + k - I)_ (n + k + e - 2) (n + k - 2) .n-I n-I n-I n-I

Remark 2.2. The present polynomials Qk,.(X) are related to the Jacobi poly-nomials Gk(p, q, x) as follows, cf. ref. I:

for R":ree + tn + 2k)

Qk .(x) = 2e+2k Gie + ten - 2), e + t, x);, r(tn) ree + 2k + I)

ree + n + 2k)Qk (x) = Gk(e+ n - I e + I x).

,. ren) rek + I) ree + k + I) "for en:

For R", they are related to the Gegenbauer polynomials C/In)(x) by

ree + I + 2k) rem)x· Gk(e+ m, e + t, x2) = c.+2k(m)(X).

. 2·+2k ree + m + 2k)

We shall refer to the polynomials Qk,.(X) as Jacobi polynomials. The fol-lowing theorem holds for both R" and en, and is a consequence of the defi-nitions and of remark 2.2.

Theorem 2.3. For each e E {O,I}, the polynomials Qo,.(x), Ql,.cX), ... , Qk,.(X)form a basis for the linear space of all polynomials of degree~ k. This basis is orthogonal on [0, I] under a suitable weightfunction. The coefficients in

'+1+.x· Q".(x) Q1,.(X)= L q.(i,j, k) Qk,O(X)

k=O

satisfy, with Kronecker (3',1,

qo(O,j,k) = (31,k; q.(i,j,O) = Q".(I) (3',J'

3. Addition formulaeIn R", with n ~ 2, let Wn denote the volume of the unit sphere IJn•

Definition 3.1. Harm (I) is the linear space of all harmonic functions on IJnwhich are represented by homogeneous polynomials of degree Iin n variables.

It is well known 9) that Harm (I) has the dimension

N=N(I)=(n+l-I)_(~+1-3) for I~I, N(O)=l.n-I n-l

BOUNDS FOR SYSTEMS OF LINES. AND JACOBI POLYNOMIALS 95*

In C", with n ~ 2, endowed with the Hermitean inner product, let Wn de-note the volume of the unit sphere Qn- Let

denote any polynomial in 2n variables, with complex coefficients, homogeneousof degree I in Yl' ... , Ym homogeneous of degree k in Zl' ••• , Zn' It is calledharmonic if it satisfies the condition

n

1=1

Definition 3.2. Harm (I, k) is the linear space of all functions S on Qn definedby

where S(y;z) is a harmonic polynomial, homogeneous in Y; zof degrees I;k.

It is well known 10) that Harm (I, k) has the dimension

_ _ (n + 1- 1) (n + k - 1)_ (n + 1- 2) (n + k - 2)N - N(/, k) - .n-l n-l n-l n-l

Theorem 3.3. For each e E {O,I}, and integer k ~ 0, the linear spacesHarm (2k+ e) in the case R"; Harm (k+ e,k) in the case C"have the dimension N = Qk .•(1). For any orthogonal basis(Sl' S2' ... , SN) of these spaces, with norm (SI) = VWm thefollowing addition formula holds:

In view of remark 2.2, this formula coincides in the real cases with the classicaladdition formula for Gegenbauer polynomials, cf. ref. 9, In the complex case,this formula is a special case of the addition formula for Jacobi polynomials,which was recently obtained by Koornwinder 10.11).

4. Characteristic matrices

From now on, we treat the real and the complex cases simultaneously. LetX denote afinite nonempty subset ofthe unit sphere Dn' of cardinality lXi =: v.For any fixed labelling of X, for e E {O,I}, for integer k ~ 0, and for any

96* P. DELSARTE. J. M. GOETHALS AND J. J. SEIDEL

orthogonal basis (SI' S2' ... , SN) with norm (S,) = VWn of Harm (2k + e),and Harm (k + e,k), respectively, we define the matrix Hk.' as follows:

Definition 4.1. The characteristic matrix H; .•, of size vxN, with N = Qk.•(l),is the matrix .

H;.•=[St(~)], ~eX, te{1,2, ... ,N}.

Thus, each column of H, .• consists of the values taken on the vectors of X bythe corresponding basis polynomial. Without loss of generality, we let Ho•o bethe all-one vector, which we denote by u. We use the following notations. Forany matrix M its conjugate transposed is denoted by M, and its norm byI!MII= (tr MM)l/2. The matrix LIk., denotes the zero matrix 0 for k =1= 1,and the unit matrix I for k = 1.

Lemma 4.2. u,..flk.o= [(~,17)0 Qk.•(I(~, 17)12)];

IIRk,« H,.oW = L I(~,17)12• Qk.•(I(~, 17)12) Q, .•(I(~,17)12);I!.IleX

Proof The first identity is a direct consequence of the addition formula oftheorem 3.3; the second follows from the first one by straightforward verifica-tion; the third one uses

We now approach the crucial theorem 4.4. For any e e {O,I} we associateto any polynomial F(x) e R[x] its expansion in the basis of the Jacobi poly-nomials Qk.•(X):

F(x) = Lik.' Qk.'(X).k=O

Definition 4.3. For any integer i ~ 0, the polynomial F(x) is called (i, e)-compatible with the set X C Qn whenever

Theorem 4.4. For any e e {O,I} and integer i ~ 0, any polynomial F(x) whichis (i, e)-compatible with a set X C Qn of cardinality v satisfies

v Q, .•(l)(F(l)-vjj.J ~ 'lJk .• I!R,.•H; .• -vLf,.kW,k=O

L ::s:;; L QI ••(I) F(l) = V Qt ••(l) F(l).~eX

BOUNDS FOR SYSTEMS OF LINES. AND JACOBI POLYNOMIALS 97*

Moreover, equality holds if and only if

Proof. Consider the sum

L := L I(~,1]W· QI,.(I(~,1])12)F(I(~, 1])12).~."eX

Since F(x) is (i, e)-compatible with X we have

On the other hand, expansion of F(x) in {Qk .•(X)} and application oftheorem4.2 yields

00

L =L L I(~, 1])12• Q, ..(I(~, 1]W) Qk .•(I(~, 1])12)fk .•k=O ~",eX

00

= LA.IIB, .• tt,.•112k=O

00

= LA.IIB, .• Hk.B - V L11.kI12+ v2 fi .•Q, .•(I),k=O

This implies both assertions.

I+J+.Lemma4.5. IIBI •• HJ .• -vL1I•JW= L q.(i,j,k)llaHk.oW,

k=l

where u is the all-one vector, and q.(i,j, k) are as in theorem 2.3.Proof. Using lemma 4.2 and theorem 2.3 we observe that

t+J+.IIBI.•HJ .•W = L q.(i,j, k) L Qk.o(I(~,1])12),

k=O ;:.lleX

lIa Hk.OW = L Qk.o(I(~, 1])12).1;.lleX

Again by 4.2 and 2.3, this readily leads to the assertion.

98* P. DELSARTE, J. M. GOETHALS AND J. J. SEIDEL

5. Special bounds for A-sets

Let A = {1X1>1X2'' .. , IXs} denote a finite set of s ~ 1 distinct real numbers,with 0 ::;;;;IXI < 1. •

Definition 5.1. A finite nonempty set Xc Dn is an A-set whenever

Theorem 5.2. For any e E {O,I}, let F(x) be a polynomial satisfying

VaeA (IX· F(IX) ::;;;0); Vk~l U«,~ 0); fo,. > 0,

for the coefficients fk,. in its expansion in Qk,.(X). Then

lxi::;;;; F(1)/fo,.·Proof Since Qo,. is a positive constant, F(x) is (0, e)-compatible with X. Nowapply theorem 4.4, and use the hypothesis.

Remark 5.3. In the case of equality the second part of theorem 4.4 implies that

VaeA (IX· F(IX)= 0), Vk~ 1 ((fk,. > 0) => (îi Hk,. = 0)).

This is useful for a discussion of A-sets achieving the bound. We shall notpursue a complete discussion in the present paper. However, we refer to sec. 7,in Mrticular theorem 7.5.

We now make theorem 5.2 explicit by special choices for ê and F(x), de-pending on A. Call A* :=A\{O}.

Definition 5.4. Type (A) equals 1 for 0 E A, and 0 for 0 ~ A.

Definition 5.5. The annihilator of A is the polynomial

nX-1X nx-IXx-· -- = -- , with e = type (A).

I-IX I-IXaeA aeA·

Thus, the annihilator of A is the polynomial of degree s - e which vanishesfor all IX E A*, and takes the value 1 for x = 1. We now apply theorem 5.2for e = type (A) and F(x) the annihilator of A. This yields a bound v(A), say,for v = lXi provided fo,. > 0 and all fk,. ~ 0 in the expansion of F(x).Table I contains the results, both for R" and for en, for s = 1, s = 2, ands = 3 with 0 E A. The validity of v ::;;;;v(A) depends on two conditions:(i) the denominator of v(A) should be positive,(ii) IX+ {J should not exceed a certain value, K say, for s = 2, 3.The results of the table are valid for all such values of 0 ::;;;;IX< 1, 0 ::;;;;{J < 1,hence also for {J = 0 and for IX = {J.

TABLE I

BOUNDS FOR SYSTEMS OF LINES. AND JACOBI POLYNOMIALS 99*

Special bounds

field A v(A) K

R,e {ex}n (1 - ex)

1-n ex

R {ex,P} n (n + 2) (1 - ex)(1 - (3) 6--

3 - (n + 2) (0: + (3)+ n (n + 2) 0: (3 n+4

n (n + 1)(1 - ex)(l - (3) 4C {ex,(3} --

2- (n + 1)(0: + (3)+ n (n + 1) 0: (3 n+2

n (n + 2) (n + 4) (1 - 0:) (1 - (3) 10R {O, 0:, (3} --

IS - 3 (n+4)(0:+ (3)+ (n+2)(n+4) ex(3 n+6

n (n + l)(n + 2)(1 - 0:)(1 - (3) 6e {O, ex,(3} --

6 - 2 (n + 2)(0: + (3)+ (n + l)(n + 2) ex(3 n+3

Remark 5.6. The annihilator of A needs not be the best choice for F(x). Thisis illustrated for s = 1 and s = 2 in the case R", Let A = {«} and B = [«, (3}with 0 ~ (3 ~ 0: < l/n. Then for B the choices e = 0 and F(x) = x - 0: satisfythe hypothesis of theorem 5.2, yielding v ~ n (1 - 0:)/(1 - no:), which is betterthan v(B) of table I, for 0 ~ (3 ~ 0: < 1/(n + 2). Conversely, in case of A-sets,v(B) with any suitable (3yields a better value than v(A) for 0: > 1/(n + 2). Thelimit value 0: = l/(n + 2) yields v(A) = -tn (n + 1) = v(B), for every (3. Infact, -in (n + 1) is the absolute bound of table 11.

Example 5.7. In the case R", A = {«}, table I yields

n 2 3 4 5 6 7 15 19 20 21 22 230:-1 4 5 9 9 9 9 25 25 25 25 25 25

v(A) 3 6 6 10 16 28 36 76 96 126 176 276,

In the cases n = 19 and n = 20, it is unknown whether the bound v(A) can beachieved. In all other cases an extremal set of equiangular lines has been realizedby a regular two-graph, cf. refs 15 and 16. Sometimes, these sets have interestingautomorphism groups: for n = 21 the unitary group prU(3, 52), for n = 22the Higman-Sims group, for n = 23 Conway's group ,3.

100* P. DELSARTE. 1. M. GOETHALS AND 1.1. SEIDEL

Example 5.8. In the case C", A = {oe}, we realize

n = 2m, oe-I = 4m -1, v(A) = 4m,

for many values of m, as follows, cf. ref. 17. Let e be a skew conference matrixof order 4m, that is, a skew matrix with elements 0 on the diagonal and ± 1elsewhere satisfying eeT = (4m-1) J. Such matrices coexist with skewHadamard matrices of order 4m. The complex matrix J + i (4m_1)-1/2 eis Hermitean positive semi-definite of rank 2m. Hence, C2m contains 4m unitvectors with Hermitean inner products ± if(4m-1)l/2.

Example 5.9. In the case C", A = {O,oe}, the following examples have beenrealized:

(n, oe-I, v(A» = (5,4,45), (9, 9, 90), (28,16,4060).

The first and second example may be obtained from the regular two-graph on276 vertices, and will be treated elsewhere. For the third example, whoseautomorphism group is Rudvalis' simple group, we refer to ref. 5.

6. Absolute bounds for A-sets

In this section we obtain upper bounds for the number of vectors of anA-set Xc Dn' depending only on the cardinality s and the type e of A, andnot on the elements oeI' oe2' ... , oes of A. Moreover, if the bound is achieved,then the elements of A turn out to be determined by n, s, e.

From the characteristic matrices Hk .e of a finite set Xc Dn we constructthe matrix

H. = [Ho ••, Hl ...... , Hç: •.•l.Since Hç .• has Qk ••(1) columns, H. has

s-.M(s, e) :=L a...(1)

k=O

columns. By use of the explicit expressions for Qk .•(1) mentioned in sec. 2,we deduce

(n + 2S-e-1)

M(s,e)= n-1 for R"; (n+s-1) (n+s-e-1)= for C",

n-1 n-1

Theorem 6.1. For any A-set Xc û; the inequality

lXi ~ M(s, e), with s = lAl, e= type (A),

holds. In the case of equality the annihilator of A is the poly-nomial

1s-.- L Qk ••(X).v k=O

BOUNDS FOR SYSTEMS OF LINES. AND IACOBI POLYNOMIALS 101*

Proof Let s = type (A), let F(x) be the annihilator of A, and let.-.F(x) = "iJk •• Qk .•(X)

k=O

be its expansion in the Jacobi polynomials Qk.•(X). Define the diagonal matrixLI. of size M(s, e) by

LI. :=/0 .• 10 tB/I .•II E9 ••• $/.- •.•1.-.,where Ik denotes the unit matrix of size Qk .•(I). By use of lemma 4.2 and theexpansion of F(x) we observe that

H. LI. H. = [(~,'YJ). F(I(~, 'YJ)12)] = I.

This proves the inequality for v = lXi, since it implies

min {v, M(s, e)} ~ rank H. ~ rank (H. LI.H.) = rank I = v.

Next, we assume the bound to be tight, that is,

v = M(s, e).

Then H. is nonsingular and LI. is positive definite. Hence all diagonal entries/k .• of LI. are positive. Since F(x) is (i, e)-compatible with X, application oftheorem 4.4 yields

°<It.. ~ I/v, for i = 0, 1, ... , s- e.

In addition, from .-. .-.v = v F(I) = v L lt.. Q,.•(I) ~ L Q,..(I) = M(s, e)

1=0 1=0

it follows that

ft .•= I/v, for i = 0, 1, ... , s - e,

which implies the assertion.

Remark 6.2. Theorem 6.1 implies that, if the bound is tight, the elements ofA are the zeros of the 'polynomial.-.

L Qk .•(X).k=O

From the defining equations for Qk••(n;x) := Qk.•(X) it can be proved, inview of remark 2.2, that

r nk~ Qk.C(n;x) = n + (2-y) (2r + e) Qr.•(n + y; x),

TABLE 11

Absolute bounds

102* P. DELSARTE. J. M. GOETHALS AND J. J. SEIDEL

with y = 2 for R" ,and y = 1 for en. This again stresses the importance ofthe Jacobi polynomials for our theory.

Table 11 contains the explicit bounds, and the accompanying annihilators,suitably normalized, for R" and for en, in the cases s = 1, s = 2, and s = 3,8 = 1.

field s 8 M(S,8) annihilator

R,e 1 1 n , I

R 1 ° e:I) (n+ 2) x-I

e 1 ° n2 (n + 1) x-I

R 2 I e:2) (n + 4) x- 3

e 2 1 ne: 1) (n+ 2) x-2

R 2 ° e:3) (n+ 4) (n + 6) x2 - 6 (n+ 4) x + 3

e 2 ° e:ly (n+ 2) (n + 3) x2 - 4 (n + 2) x + 2

R 3 1 e:4) (n+ 6)(n + 8) x2 -10 (n + 6) x + 15

e 3 1 r:1) (n:2) (n+ 3)(n + 4) x2 - 6 (n + 3) x + 6

Example 6.3. For R", the following realizations are known, cf. refs 4 and 15:

n=7, M=28 , A = {t} , Aut = Sp(6, 2);

n = 23, M=276 , A = Us} , Aut = Con-S;

n= 8 , M= 120 , A = {O,t} , Aut = WeEs) ;n = 23, M= 2300 , A = {O,H , Aut = Con '2;

n = 24, M= 98280, A = {O,t, 116}' Aut = Con ,1.

BOUNDS FOR SYSTEMS OF LINES, AND JACOBI POLYNOMIALS 103*

Example 6.4. For en, the following realizations are known:

n = 2, M = 4, A = {t} n = 3, M = 9 , A = {!}n=4, M=40, A={O,t}; n=6, M=126, A={O,t}.

Details about these line systems, for which we refer to Coxeter 6) andMitchell !"), will appear elsewhere.

7. Properties of extremal A-sets

In this section we exhibit some algebraic and combinatorial properties ofA-sets achieving the bounds of theorems 5.2 and 6.1. For similar results in thetheory of t-designs, we refer to Cameron 3) and Delsarte 8).For A = {(Xl' ••• , (Xs}, let Xc Qn be an A-set of cardinality v. For any

labelling of X, for any e E {O,I} and j e {I, 2, ... , s}, the square matrix DJ .•of order v is defined by its elements

DJ ..(~, rJ) := (~,17)" if I(~,17)jz= (XJ; = ° otherwise,where ~ and 17run through X. Next, we define the linear spaces Ao and Al overthe field C as follows:

A. := (1, Dl ••, D2 .•, ... , D,.•).

Since Dl,l = ° for (Xl = 0, we have

dim A. = s + 1- ~ e, where ~= type (A).

We are interested in conditions for A. to be an algebra, that is, to be closedunder matrix multiplication. Among the reasons for this interest are the com-binatorial properties of such algebras. Rather than going into details, wemention the following examples.

Example 7.1. For an A-set X two distinct elements ~,17EX are called ithassociates whenever I(~,17)12= (XI holds. This definition yields an associationscheme if and only if Ao is an algebra, cf. ref. 2.

Example 7.2. In the real case R", an A-set X with s = 1, type (A) = 0, forwhich Al is an algebra, corresponds to a regular two-graph, cf. refs 15 and 16.

For e E {O,I} and X C Dm let Ho.•, Hl,.' H2.•, ... denote the characteristicmatrices defined in 4.1. We construct the following matrices of order v = lXi,for i = 0, 1, 2, ... :

JI .e := V-I Hl .e Hl .•'

Lemma 7.3. Ifa Hk•o = 0, for k = 1,2, ... , 2d + e, then Jo .•, Jl •• , ••• , Jd ••

are idempotent and pairwise orthogonal.Proof By application of lemma 4.5 we have

Hl .e HJ •• = V LlI.i> for °~ i +j ~ 2d,which proves the assertion.

104* P. DELSARTE, J. M. GOETHALS AND J. J. SEIDEL

Theorem 7.4. If for an A-set X C Dm with 8 = lAl, (j = type (A), we have

il Hk,o = 0, for 1::S;;;;k :::;;2 (8-1- e (j) + e,

then A. is a commutative algebra.Proof. Let X be an A-set with A = {(Xl' (X2' ••• , (Xs}. By the first formula oflemma 4.2 the matrices 1". satisfy

vI". = Q".(I) I +L Q',.((XJ) DJ,.,J=l

hence belong to A., for all integers i ~ O.Put

d := 8 - 1 - s è3 = dim A. - 2.

If we can prove that the d + 2 matrices

form a basis for A., then the assertion follows by application of lemma 7.3.So suppose that these matrices are linearly dependent with coefficientsCer;, Co, Cl> ••• , Cd' with at least one 0 =1= c, E {co, Cl> ••• , Cd}' Then, by sub-stitution, we have

But I, Dl..' D2, ..••• , Ds,. are linearly independent, except for Dl,. = 0 inthe case (Xl = 0, e (j = 1. Hence the nonzero polynomial

Co Qo,c(x) + Cl Q1,.(X) + ... + Cd Qd,.(X),

of degree s; d, vanishes for all (XJ E A, except possibly for (Xl = 0 and s (j = 1.This is impossible by d = 8 - 1 - e (j. Now the theorem is proved.

Theorem 7.5. Let the annihilator F(x) of A, with 8 = lAl, (j = type (A), satis-

fy ft,6 > 0 for i = 0, 1, ... ,8- (j. Let

10,6=/1.6 = ...=L». for t:= max {O,8+ e-2 (1+ s (3)}.

If X C Qn is an A -set achieving the special bound

VlO,6 = F(1) = 1,

then A. is a commutative algebra.Proof. Let the polynomial Xt+6 F(x) have the Jacobi expansion

S+l

xl+6 F(x) = L Ck Qk,O(X).k=O

BOUNDS FOR SYSTEMS OF LINES. AND JACOBI POLYNOMIALS 105*

The coefficients Ck may be expressed in terms of the given fi.6' and it readilyfollows from the recurrences of sec. 2 that

Co = fO.6' Ck > 0, for k = 0, 1, ... , s + t.

Since xt+6 F(x) is (0, Oj-compatible with X, theorem 4.4 may be applied. To-gether with Ck > 0 the condition F(1) - v Co = 0 yields

11Hk•O = 0, for 1 ~ k ~ s + t.

By theorem 7.4 this proves the theorem, since

s + t ~ 2 (s - 1 - 8 b) + 8.

Theorem 7.6. If an A-set X achieves the absolute bound v = M(s, b), then ADand Al are commutative algebras.

Proof. According to theorem 6.1, the hypothesis implies fO.6 = fl.6 = ...=J.-6.6 = I/v. Hence the desired result directly follows from theorem 7.5.

Remark 7.7. Theorems 7.5 and 7.6 yield information about the extremal A-sets, and provide necessary conditions for their existence.

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