Discrete Mathematics 114 (1993) 237-252 North-Holland
237
(s, r; ,u)-nets and alternating forms
graphs
Tayuan Huang
Department of Applied Muthematics, National Chiao-Tung University. Hsin-Chu 30050, Taiwan. China
Monique
Laurent
LIENS, Ecole Normale Suptrieure 45 rue d’Ulm. 75230 Paris cedex 05, France Received 20 January 1989
Revised 22 February 1989
Abstract
Huang, T. and M. Laurent, (s,r;n)-nets and alternating forms graphs, Discrete Mathematics 114 (1993) 2377252.
The equivalence between Bruck nets and mutually orthogonal latin squares is extended to (s, r; u)- nets and mutually orthogonal quasi frequency squares. We investigate geometries arising from classical forms such as bilinear forms, alternating bilinear forms, hermitian forms and symmetric forms and show that (s, r;p)-nets provide the right building blocks for each of these geometries with suitable values of p. Toward the goal of geometric classification of distance-regular graphs, the local structure of the case of alternating forms graphs is stressed.
1. Introduction
The structure of (s, r; &nets includes Bruck nets as the special case of ,u = 1 and their duals are transversal designs TD, [r, s] introduced by Hanani [6]. Indeed, (s, r; ,u)-nets are equivalent to affine designs S,( 1, sp, s*p) and to orthogonal arrays OA,(s, r), and in this language they have been studied since around 1945. A survey on the geometric and group-theoretic aspects of (s, r; p)-nets can be found in [ 111, where problems concerning existence, completion and geometric configurations are emphasized.
In Section 2, we recall the notion of (s, r; CL)-nets, the procedure of ‘inflation’ is used for constructing (s, r;p)-nets from existing (s, r; I)-nets. In Section 3, the notion of quasi frequency squares is introduced and then we prove the equivalence between (s, r; n)-nets and sets of mutually orthogonal quasi frequency squares, which includes the well-known relationship between Bruck nets and sets of mutually orthogonal latin
Correspondence to: Tayuan Huang, Department of Applied Mathematics, National Chiao-Tung University, Hsin-chu 30050, Taiwan, China.
238 T. Huang. M. Laurent
squares as a special case. After reviewing how nets (with p= 1) provide the right building blocks for the lower semilattice 6p4( I’, W) [4] by using Sprague’s [14] result on the characterization of d-dimensional nets, we study in Section 4 the geometries associated with classical forms (alternating bilinear forms, symmetric forms, hermitian forms and bilinear forms) and show how their local structures involve (s, r; p)-nets (with ~22). We also emphasize the relationships between these geometries and the association schemes of affine type carried by each family of the above classical forms. In the final section, we further investigate the geometric properties of the association schemes defined over alternating bilinear forms, hence covering the initial step toward the problem of characterization of their graphs by their intersection arrays.
2. (s, r;p)-nets
In this section, first we xshall recall the notion of (s, r; p)-nets, and then a specified class of (s, r; p)-nets which can be obtained from given (s, r; 1)-nets by the procedure of ‘inflation’ will be studied. The diagram
[lip o-o
is introduced for the class of duals of (s, r; y)-nets.
A finite incidence structure Ii’= (9, iii?‘, E) is called a (s, r; p)-net ofmultiplicity p if the block set g can be partitioned into r (r2 3) block classes gl,aZ, . . ..g’. such that
(Nl) the blocks of each block class %?i form a partition of 9, (N2) any two blocks from distinct block classes meet at p points,
\J3) one of the block classes consists of s blocks.
Since r b 3, it follows that each block class al, gZ, . . . , W, consists of s blocks, each block in 98 consists of sp points, each point lies on exactly r blocks and, hence, 15??l=s2/l.
For an (s, r; p)-net, it is known that [l l] r d (s2p- l)/(s - l), with equality holding if and only if it is an affine 2-design. Indeed, if r = (s2p - l)/(s - l), then any two points are on precisely i =(sp- l)/(s - 1) blocks. The sets T(r, p) of integers s for which (s,r;p)-nets exist were investigated by Hanani [6]; in particular, he showed that SE T(7, p) for every s > 1 and every p 2 2. Recently, Ray-Chaudhuri and Singhi [ 131 have shown that, for given r and s, there exists an (s, r; p)-net whenever ~1 is sufficiently large. (s, r; 1)-nets are simply the well-studied classical Bruck nets; for example, the existence of an (s, r; 1)-net is equivalent to the existence of each of the following structures:
(1) r - 2 mutually orthogonal latin squares of order s, (2) an orthogonal array OA(s, r),
(3) an affine design S,(l, s, s*), (4) a transversal design TD[r, s].
The intersections among blocks would shed light on the structures of (s, r; p)-nets, in particular when p >/ 2. Let n = (9, B, E) be a (s, r; p)-net; for any two blocks B and B’ of 17, we write
B//B'
if they are in the same class (soBnB' =8),
andB# B'
otherwise (so(s, r; p)-nets and alternating forms graphs 239
I&M =p). For BES?~, and j # i, let
bj(B)={BAB’IB’EBj},
6(B)= U &j(B).
lsj<r j#i
Clearly, each bj (B), 1 <j d r and i #j, forms a partition of B. For 1 < i, j < r, i #j, let
&ij= {BnB’IBE9Yi, B’E~j},
and
d= U eij.
lai,j<r iij
Then bij forms a partition of 9, and elements of &G(,;) are called lines. For distinct pairs (i, j) and (2, j’), where 1 <i, i’, j, j’ dr, i #j, i’ #j’, the two partitions bij and Bi,j, of the point set 9 do not necessarily coincide. An interesting special case is when all such partitions coincide, i.e., dij = di,j, for all 1 < i, j, i’, j’ <r, with i #j, i’ Zj’. This happens exactly when the following condition (*) holds:
(*) For any two intersecting blocks B and B’ (say BEB~, B’E~~), there exist blocks Bi~~i, 1 <i<r, such that BnB’= nl<i,, Bi.
On the other hand, we may define a relation N on 9’ in such a way that, for any x, y~9, x ‘Y y if and only if x, y~Z?nBl for some distinct blocks B, B’E~. Under condition (*), the relation ‘v is clearly an equivalence relation on 9, with & as its family of equivalent classes. The above observations are summarized in the following theorem.
Theorem 2.1. An (s, r; p)-net Z7=(9, a,~) satisfies condition (*) $ and only ij’ the
associated incidence structure Zl/- =(a,%?, G) is an (s, r; 1)-net.
The following procedure of inflation will provide us (s, r; ,u)-nets satisfying condition (*) from existing (s, r; 1)-nets. Let 17 = (9, W, E) be a given (s, r; 1)-net and y be a given positive integer. Let
@=1(x, i)lxeP and 1 <i,<p},
g={(x,i)(x~B and l<i<@>,
where BEG, and
&?={(B(B&}.
Then one says that the incidence structure I? =(@, &,E) is obtained from I7 = (9, &I, E) by inflation. The following can be easily verified.
Theorem 2.2. (1) The incidence structure fi = (.@‘, &“, E) is a (s, r; ,a)-net satisfying condi-
tion (*).
(2) Zf ZZ = (.9,93, E) is a (s, r; p)-net satisfying (*), then Zi’ = (9, a’, E) can be obtained from the (s, r; I)-net ZZ/- =(b,@, 5) by inflation.
240 T. Huang, M. Laurent
Certainly, (s, r; p)-nets II = (9,93, E) satisfying (*) possess tighter structures than general (s, r; p)-nets do. For instance, let ri (x) = (z 1 ZEN’ and x, z are in a common block},
(1) If BEB and x is not in B, then T1(x)nB consists of r- 1 pairwise disjoint lines, and Ir,(x)nBI=p(r-1).
(2) If x, YE? are not collinear, then the blocks containing x or y can be indexed as Bi and B;, where 1 <i, j<r such that Bi# B; whenever i #j and Bi//BI, 1 <i<r.
For a prime power q, a specific class of (q”-I, q + 1; q)-nets satisfying (*) and related to alternating bilinear forms defined over finite dimensional vector spaces over finite field GF(q) will be studied in Section 5. In order to classify d-injection geometries, the diagram
[dl o-o
was used by Deza and Laurent [S], for a class of rank-2 geometries, i.e., d-transversal planes, where d 3 1 is an integer. When d = 1, any incidence geometry belonging to
[ll (11,
o-o (i.e., o- 0)
is simply the dual of an (s, r; 1)-net, i.e., TD (r, s). Following this line, the class of the duals of (s,r;p)-nets will be denoted by the diagram
in the rest of this paper. In Section 4, we shall show that geometries associated with classical association schemes of affine type belong to diagram
Cl1 o-_o-0 . . . . . o-_-o PO
with suitable choices of p. The reader is referred to [l] for the details of association schemes and to [2] for the details of diagram geometries.
3. (s, u; p)-nets and quasi frequency squares
In this section, relaxing the condition required for frequency squares [12], we shall introduce the notion of quasi frequency squares, which includes latin squares and frequency squares as special cases. Moreover, the well-known equivalence between (s, r; 1)-nets and sets of mutually orthogonal latin squares can be generalized to (s,r;p)-nets and sets of mutually orthogonal quasi frequency squares.
Afrequencysquare F(n;pl,p2,..., pL,) of order n (FS for short) is an n x n array with entriesfromtheset[l,s]={1,2,..., s} with the property that each symbol YES occurs exactly ,ui times in each row and each column. Clearly, n=pl +p2 + . . . +ps and an F(n; l,l, . . . . 1) frequency square of order n is simply a latin square of order n. Two frequency squares F (n; pl, pz, . . . , ps,) and F (n; vl, v2, . . . , v,,) are said to be orthogonal if each ordered pair (i, j) of symbols occurs exactly ~ivj times for all i~[l,si] and
(s, r; p)-nets and alternating firms graphs 241
j~[l,sJ when the square F2 is superimposed on the square F1. A set {F,, FZ, . . . . F,} of r > 2 frequency squares is said to be mutually orthogonal if Fi is orthogonal to Fj when- ever i #j. In what follows, we shall consider only frequency squares with ,u~ = . = ps = p; a frequency square of this type is called a frequency square F (s, p) of frequency p on s symbols. Such frequency squares can easily be constructed from latin squares by the following inflation procedure: Take a latin square L of order s, replace each point (x, y)~ [l, s]’ by the p x p matrix whose entries are all equal to L(x, y); then what we obtain is a ps x ,US matrix F which is obviously a FS of frequency ,D on s symbols.
We need some more notations to introduce the notion of quasi frequency squares. A partition .d of [ 1, n]’ is called a row partition if its classes are row-closed, i.e., for any class AE,~, (a,~)~,4 for all ye[l,n] whenever (a,b)~A for some b. Obviously, a row partition of [l, n12 is uniquely determined by a partition of [I, n]. Similarly, a column partition is a partition %? of [ 1, n]’ whose classes are column-closed. For a given n = ps, with ,u,s~ 1, let X, denote the partition of [l, n] into s classes {[(k- l)p+ l,kp] 1 1 dkbs}, where [(k- l)p+ l,kp] is the set of all integers between (k- l)p+ 1 and k,u (included), and let J.@‘,,, VYO denote the row partition and the column partition, respective- ly, of [l, n12 determined by 3,. Hence, both JZI~ and %‘0 consist of s classes, each class consists of ~1’s points, and any two classes AEG!~, CM?, intersect in p2 points.
A quasifrequency square of frequency p on s symbols (QFS for short) is an n x n array, n = ,us, with the property that each symbol ie [ 1, s] occurs exactly ,M’ times in each class A of J310 and in each class C of VO. Clearly, any frequency square F(s,p) is a quasi frequency square, and a QFS F(s, 1) is simply a latin square of order s. Orthogonality between QFS can be defined similarly as before, i.e., each ordered pair (i,j) of symbols occurs exactly pL2 times for i, j~[ 1, s] when one QFS is superimposed over another.
Let {F,, . . . , F,} be r mutually orthogonal QFS of frequency ,U on s symbols [l, s]. Let n=ps; we set P=[l,n12, .%?~={[i,a]Il<a<s}, l,<i<r: where
Ci,xl={(X,Y)E~lFF,(x,y)=a},
and B= u Bi. i,<iQrSome easy observations are as follows:
(1) Each block in 93 consists of ,u2s points, and each family gi consists of s blocks which form a partition of 9.
(2) Any two blocks from distinct families Bi, &Ij intersect in p2 points (by the orthogonality among (F, , . , . , F,}). Hence, the incidence structure
242 T. Huang. M. Laurent
(3) By the assumption that each FL is a QFS, one can extend the above QFS-net by adding two more block classes do and wO, i.e.,
is a (s, r + 2; p2)-net.
Equivalence indeed holds between sets of mutually orthogonal QFS and nets of multiplicity ,LL’ as shown in the next theorem.
Theorem 3.1. The following assertions are equivalent:
(1) there exist r mutually orthogonal quasi frequency squares of frequency p on
s symbols,
(2) there exists a (s, r + 2; p2)-net.
Proof. It remains to show that (2) implies (1). Let Xl = (9, 99, E) be a (s, r + 2, p2)-net and let d, +Z, 2 1, . . . ,93,. denote its r + 2 block classes. Then the point set 9 can be partitioned into the s2 sets AnC of size p2, where AE& and CM. It is easy to see that this enables us to represent 9’ as [l, n]‘, n=ps, in such a way that d, ‘$? are the row partition &‘, and the column partition wO, respectively. For 1 <j<r, let
We define an n x n matrix Fj, with entries in (1,2, . . ..s}. by setting Fj(X, y)= k if and only if (x, y)~Bp). Since 17 is a (s, r + 2; p)-net,
for all AE~, CE%‘,B:~)EB~, and jB:“nBp’I =p’ whenever i#j. It follows that each Fj is a QFS of frequency /J and any two Fi, Fj are orthogonal. Hence, { F1, . . . , F,.} is a set
of r mutually orthogonal QFS of frequency p on s symbols. 0
Corollary 3.2. If there exist r mutually orthogonal frequency squares of frequency p on
s symbols, then there exists a (s, r + 2; p2)-net.
We now turn to the question of determining whether it is possible to derive a set of
r mutually orthogonal FS from a (s, r + 2; p2)-net. In fact, as shown in the following
example, this is not true in general.
Example33. Let~=[1,6]2,LQZo={A1,A3,A5},~~={C1,C3,C5}and~==(B,,Bz,B,},
where Ai={(i,y), (i+l,y)ll<y<6}, Ci={(y,i),(y,i+l)lldy~6) for i=l, 3, 5 and Bi = ((1,1X (29 3)> (2,5), (2,6), (3, l), (4,3), (4,4), (43% (5,1X (5,2), (5,5), (63))) J-b = {U > 3 (1,5), (2, l), (2,4), (3,4), (3,6), (492) (4,5), (5,3), (5,6), (6% (6,4)},
(s, r; p)-nets and alternating ,forms graphs 243
Then Zl= (9, dou%?,-,u~, E) forms a (3,3; 4)-net. Clearly, 17 correspond to a QFS of frequency 2 on 3 symbols, but it is certainly not a frequency square as shown in the following diagram, where value k, 1 d k < 3, appears at position (i, j), indicating that (i,j)c&. 123323 231211 133232 321121 112312 321233
Quite naturally, if we require that a (s, r+2;$)-net satisfies (*), mentioned in Section 2, i.e., be the inflation of some (s, r+2; 1)-net, then there exists a set of r mutually orthogonal FS which, in fact, are the inflation of some mutually orthogonal latin squares.
Theorem 3.4. If there exists a (s, r + 2; p2)-net which satisfies condition (*), then there exist r mutually orthogonal frequency squares of frequency p on s symbols.
Proof. Following the notation used in the proof of Theorem 3.1, it only remains to verify that, for all 1 <j < r and BESTS, B contains exactly /A elements in each row and in each column. Fix AES, B= B~‘E~j. By condition (*), we can find CM such that AnB= AnBnC= AnC; this implies that Fj(X, y)=k for all (x, y)~Anc and, hence, Fj is indeed the inflation of a latin square of order s. 0
Of course, we would like to obtain a set of mutually orthogonal frequency squares from a (s, r +2; p2)-net by using conditions milder than (*). One step towards this direction is to observe that, for a given set of r mutually orthogonal FS, one can sometimes derive a (s, r’;,u2)-net with r’>r+ 3, i.e., one can define additional block classes besides the families JzZ~, go, 9if1, . . . , 9i?r constructed so far. This can be done if one can define row or column partitions other than JX?,, and +ZO. Recall that 3, denotes the initial partition of [l, n] which determines do, %?,,; suppose that Y1 #JJ,, is another partition of [l,n] into s subsets of equal size p and ZZ’,, 59, denote its associated row and column partitions. We can extend the (s, r + 2; p2)-net obtained from Corollary 3.2 by adding the two additional block classes JzZ,, VI if and only if ~AOnAl~=~COnCl~=p2 for AiG&r and Ci&‘I, i=O, 1, i.e., IZOnZ,I=p/s for Z,E~~ and l,~4,; hence, s must be a divisor of ,D. Therefore, no additional row or column partition of [ 1, n]* can be added to the list { dO, %YO, gI, . . . , B, > to form a (s, r’; p)-net with r’3 r + 3 whenever s is not a divisor of p and, in particular, if ,LL < s.
We now assume that s is a divisor of p, ,u=/Is, with 3.3 1. Based on the above observations, two remarks are in order:
(1) Suppose ~33 and there exists a (s,p;;l)-net Z7=([l,ps], U14i4pYi,~) on [l,,~s] with block classes LJ~, Y2, . . . , 4,; so, each 9; is a partition of [l, ps] into
244 T. Huang. M. Laurent
s equal parts. For i~[l,p], let pi, %‘i denote the row and the column partition of [l,ps12, respectively, determined by <ft. Then
no=([l>ps]2, (,;<,+J(l;<P+)
is a (s,2p;p2)-net on [1,ps12, called the grid net determined by Zl. (2) The (s, r; p2)-net
&FS =
(
Cl,w12, u
gi>E) laisrdetermined by the set of I mutually orthogonal frequency squares is always extendible by the grid net Ilo, i.e.,
is a (s, r + 2p; p2)-net. Therefore,
(l~~~i)u(l~~~i)U(l~~ldi)~E)
the next theorem follows immediately.Theorem 3.5. Let ,tt = Es and p > 3 be the largest integer for which there exists a (s, p; A)-
net on [l, ps]. If there exists a set of r mutually orthogonal frequency squares of frequency p on s symbols, then there exists a (s, r + 2p; p2)-net which is an extension of the
grid net associated with the given (s, p; A)-net.
Moreover, if there exists a (s, s + 1; 1)-net, i.e., an affine plane of order s, and p = s, then Theorem 3.5 can be strengthened as Theorem 3.6.
Theorem 3.6. Zf there exists a (s, s+ 1; 1)-net, then the following assertions are
equivalent:
(1) There exists a set of r mutually orthogonal frequency squares of frequency s on
s symbols.
(2) There exists a (s, r + 2 (s + 1); s’)-net which is an extension of the grid net asso- ciated with the given (s, s + 1; 1)-net.
Proof. It remains to show that (2) implies (1). Let Y,,,Y1, . . ..Ya. denote the block classes of the given (s, s + 1; 1)-net defined on [l, s’]. By assumption, the block classes of the (s, r + 2(s + 1); s2)-net n can be denoted by do, dl, . . . , ds, Vo, VI, . . . , ‘GfTs, and 98 1 ,..., SYSinsuchawaythat&O,dl ,..., JG!, are the row partitions and 9Z0, %?I) . . . , gs are the column partitions of [l,s212 determined by X0,9,, . ...9,. Let F,,F2, . . ..P. be the set of r mutually orthogonal quasi frequency squares obtained from 9S1, . . . ,93, by Theorem 3.1. To show that each Fj is indeed a frequency square is equivalent to showing that each BETSY contains exactly s elements in each row and in each column. Let BE&?i and C(i be the number of elements of B occurring in the ith row of [l,s212; hence, ~xi + . . . +aS2=s3. Next we claim that Q=S for all 1 $i<s2.
(s, r; p)-nets and alternating forms graphs 245
Let Z0~90, . . . . I,E.~~ be the s + 1 blocks of the given (s, s + 1; 1)-net which contain element 1 of [ 1, s2]; then the sets {l}, I0 - (11, . . . , I, - {l} form a partition of [ 1, s2]. Let A,~~r9i be the block of n formed by the rows indexed by Zi for 0 < i<s. Since Zl is a (s, r + 2(s + 1); s2)-net, we have that 1 BnA,I = s2, i.e., Cjsli cCj= s2 for 0 <ids. Sum- ming the above s+ 1 equations, we obtain that (s+ l)cr, +C14j,<s2 Mj-cC, =s’(s+ l), and, since x1 ~ jGs2 Qj =s3, it follows that a1 = s The other cases Cli = s, 2 < i < s2, can be proved similarly. 0
4. (s,r;,u)-nets and association schemes of affine type
In this section, we shall introduce some geometries related to classical forms and then show that their local structures can be described as the dual of (s, r; p)-nets with suitable parameters.
4.1. The lower semilattice -ri”,(U, V).
First, we shall concentrate on the lower semilattice $P4(U, V) of bilinear forms (p= 1). Let U, V be vector spaces of dimensions d and IZ, respectively, (d<n) over a finite field GF(q), and U + V be the direct sum of U and V. Let
Pi = {A 1 A c U + V is an i-subspace and An V is trivial},
where O< idd. Note that the condition An V= (0) is equivalent to that of dim (rci(A))=dim(A), where x1(A) is the projection of A onto the first summand U. Furthermore, if AE_!?~, then dim (xi(A)) = dim (U) and, hence, there exists a linear transformationf: U -+ V such that A = {(x, f(x))) XE U >. It follows that, with respect to fixed bases of U and V, the set yd of roofs of the geometry
can be identified with the set Md x ,,(GF(q)) of all d x it matrices over GF(q). Similarly, each other level pi can be identified with [y] x Mi,,(GF(q)), 1 <i<d, where [y] denotes the family of all i-dimensional subspaces of U. In other words,
d;pi={(.L
y)l YECYl
and f: Y-+V is linear}. For (fT Y) and (g,Z)~uf,~ P’i, we define (f; Y) <(g, Z) if and only if YcZ and the restriction gly of g on Y is identical withf; (f; Y) and (gTZ)EUo<i<d Pi are called comparable if either (f; Y)<(g, Z) or (%Z)d(J Y). Clearly, IJObiCd c5?i is the lower semilattice LZ,(U, V) introduced by Delsarte [4]. Any flag F of type (0, 1,2, . . . . d -2) in the geometry TP=(_!%?~,~?~, . . ..P’J contains a unique (fdm2, Ud_2)~9d_2, where U,_,E[,!,] and f: Ud_2-+ V is a linear transformation. The residue Res(9) of 9 in the geometry _P=(04p0,dp1, . . . . 9J is defined to be Pug!, where g={(f; A)IAE[~!‘~], f:A-+V is linear, (fd-2,Ud-2)<(f;A)), and S?={(g,U)Ig:U-+V is linear,246 T. Huang, M. Laurent
Theorem 4.1
(Sprague
[14]). (1) The incidence structure Il =(P’, !B, E) associatedwith the residue
Res (9) of
a jag 9 of type (0, 1,2, . . . , d- 2) in the geometry 9 =(~po,~l, . . . . ~3’~) belongs to the diagram
[II o-o.
More spec$cally, the dual of
Res
(F) is a (q”, q +1; l)-net.
(2) The geometry d;p = (ZO, $P1, . . . , ~3’~) belongs to the diagram
111 0 n . . . o-o-o.
Any incidence structure Zl isomorphic
to the semilinear iocidence structure
(dpd, Yipd_
1, 2)is called an (n,
q;d)-attenuated space [14], or a d-attenuated space in
short. These are examples of the following specific class of incidence structure with
(s,r;
1)-nets as their planes. A d-dimensional net is a connected semilinear incidence
structure Il such that the following conditions hold:
(Dl) every plane is a
(s,r;1)-net,
(D2) the intersection of two subspaces is connected,
(D3) if two planes in a 3-space of Il have a point in common, then they have
a second point in common, and
(D4) the minimum number of points which generate Il is
d+1.
Sprague [14] proved that every finite d-dimensional net
(d B3) is an (n,
q; d)at-
tenuated space for some finite field GF(q). Sprague also characterized d-dimensional
nets as the duals of those incidence structures belonging to the diagram in Theorem 4.1.
As mentioned before, the set 6pd of roofs of the geometry Y = (YO, Y1,
. . . , Yd)can
. .
be identified with the set
Md x ,, (GF(q))of all
d x nmatrices over GF(q). Furthermore,
it is worth mentioning here that it also carries the structure of (P&Q)-association
schemes and, hence, distance-regular graphs. Set
LRi={(A,B)IA,BEMd..(GF(q)),
rank(A-B)=i},
Odi<d.Then
(i) LRO is the diagonal of
(Md x ,, (GF(q)))2,and {LRi
IO < i < d}forms a partition of
(Mdxn(GF(q)))2,
(ii) the transpose LRT of LRi is identical with LRi, 06
i <d.(iii) if (A,B)ELR~, then
({CICEM~~~(GF(q)), (A, C)ELRi and (C,B)ELRj}I is
a constant Pz which is independent of the choice of A and B.
In other words,
(Mdx n (GF(q)),(LRi
( 0 < i < d})forms a symmetric association
scheme of
dclasses. The reader is referred to [l] for more details about association
schemes. Furthermore,
(Mdxn(GF(q)),LR1) turns out to be a distance-regular graph
with LR1 as its edge set, denoted by
H,(d, n),and, indeed, a distance-transitive graph
of diameter
d.The d-shadow ad(g,A) of (g,A)EY, is defined to be {(f, U)l(f, U)EZ~ and
(g, A)b(f, U)}. It is easy to see that ad(g, A) is a
(0, 1,. . ..d-r}-clique
in the
graph
H,(d, n),i.e., rank(A
-B)dd -rfor A, BEcI(g, A). Indeed, Huang [9] proved
(s, r; p)-nets and alternating forms graphs 247
that each maximum (0, 1, . . . . d-r}-clique of H&d, n) is of the above form as d- shadow of some (g, A)E~?~ whenever II >d + 1 and (n, q) #(d + 1,2). In particular,
rad(Y,A)/(g,A)EYd-l}~~ d can be identified with the set of all maximum cliques
of H,(d, n).
Toward the program of classifying distance-regular graphs, Huang [8] proved that the above-mentioned distance-regular graph H,(d, n) defined on _JZd = Md X ,(GF(q)) is uniquely determined by its intersection array, subject to some extra conditions by using Sprague’s characterization of d-dimensional nets. These extra conditions were modified by Cuypers [3] recently. Following similar approaches, Yokoyama [16] proved similar results in the context of distance-transitive graphs which includes even the square case d =n.
4.2. Geometries for classical ,forms
In the second half of this section, we concentrate on square matrices as well as some examples of (s, r; p)-nets with ,u > 2. Let V be a vector space of dimension n over a finite
field GF(q), where q=p” is a prime power, Us V be a subspace of dimension i. Let Bil( U), Alt(U), Her(U) and Sym(U) be, respectively, the set of all bilinear forms, alternating bilinear forms, hermitian forms and symmetric forms defined on U. We assume that p #2 in the case of alternating forms and m=2r is even in the case of hermitian forms. Then Bil(U), Alt(U), Sym(U) are vector spaces of dimensions i2,
i(i - 1)/2, i(i + 1)/2, respectively, over GF(q), and Her(U) is a vector space of dimension
i2 over GF(p’). Bil(U), Alt(U), Sym(U) and Her(U) will be denoted by Bil(i, q), Alt(i, q), Sym(i, q) and Her(i, q), respectively when there is no confusion. Let
&i={(,fi U)l UE[~] andfEAlt(U)}, Xi={(f; U)l UE[~] and fEHer(U)), ypi={(L U)l UECYI andfESym(U)}, and
si={(f; U)/UE[~] andfGBil(U)},
where 0 d i 6 n. Let .r4 denote the geometry (JzZ~, dl, . , d,) and the other geometries Y, W and X are defined similarly.
As mentioned for the geometry 9=(_90,-44, . . . . 9,) the notions of comparability, flags and residues can be similarly defined for geometries &‘, 9, &? and Z’. For example, for the geometry d=(&o,.r4,,&2, . . . . xZ,,), (L U)<(g, W) if and only if
U c W and the restriction gla of g on U is identical with f whenever (f; U)E~~,
(g, W)E&j. Any (,f; U), (g, W)E~iU~dj are comparable if either (fT U)<(g, W) or
(g, W)<(,f; U). The residue of a flag 9 of type { 1,2, . . . . n-2) is uniquely determined by some (.fn-2, Un-2)~d”-Z, where U,_ 2 E V is a subspace of dimension n - 2
and fn_ ,~Alt (U,_ 2); more specifically, the residue Res(P) of the flag 9 is defined to be 9’u%?, where P={(J U)l UE[,!,], fEAlt(U), (fne2, U,_,)<(,J U)} and g = {(g, V) I gEAlt( V), ( fn _ 2, U, _ 2) < (g, V)}. Elements in P and 98 are called points
248 T. Huang, M. Laurent
and blocks, respectively. With respect to the induced incidence relation, we shall show that the incidence structure I7 = (P’,B’, E) associated with the residue Res(S) of a flag 9 of type (0, l,..., n-2$ in the geometries &‘, J?, 9 and 99 belongs to the diagram
VI@ o-o
with a suitable choice of p for each case.
Theorem 4.2. (1) The residue Res(9) of a fiag F of type (0, 1,2, . . . , n-2) of the
geometries d, H”, Y and 99 belongs to the diagram
[114' o--o,
where i = 1,2,3 and 4, respectively. More specifically, the dual of the residue Res(9) in the geometry & is a (q”-‘, q + 1; q)-net satisfying (*).
(2) The geometries d, A?“, Y and .&I belong to the diagram
[llqf o-o-o ... o-o-o
where i= 1,2,3 and 4, respectively.
Proof. Only the proof for the geometry d =(dO, -Qz,, . . . , d,,) is given. Similar argu- ments work for the other geometries X, Y and a’. Consider the partition
{{(_A
U)lf~AW)
and
flun_2=fn-2j
I
UE
[
1.yl , with U,_,EU)
of the point set of Res(9). Obviously, any block (J; V) is incident with exactly one point of each of the above classes.
For two distinct points (f, U),(f’, U’) f rom distinct classes, UnU’= Un_2 and fIun_Z=fl U”_, =fn_ 2. Fix a base
{vr
,
. . . , v,> of V such that ( vj, . . . , v,} is a base ofUn_2, v~EU-U~-~ and v~EU’--U~_~; those blocks (h, V) which go through (L U) and (f’, U’) can be uniquely represented by matrices of the form
- 0 CI ct3 ...
a,-
--cI 0a;
..’a:,
-a3 -IX; >
L-2
-cI, -cc;
where bj, c$, 3 <j< n, are uniquely determined by fnm2, f and
f ',
respectively. Since there are q choices for LX= h(vI, v2), it follows that there are q blocks which go through both (J; U) and(f',
U’), as required. 0(s, r; p)-nets and alternating ,forms graphs 249
With respect to a fixed base of V, each member in the sets of roofs B(n, q), A(n, q), H(n, q), S(n, q) of the geometries 8, JJ, X and Y, respectively, can be expressed as an n x n matrix, n x n antisymmetric matrix with zero diagonal, n x n hermitian matrix and an n x n symmetric matrix, respectively. Furthermore, each set of roofs also carries the structure of (P&Q)-association scheme, as we mentioned before for -Y,(U, W). Let
ARi=((A,B)IA,B~Alt(n,q) and rank(A-B)=2i},
SRi=((A,B)IA,BESym(n,q) and rank(A-B)=2i-1, 2i}, where 0 d i < [n/2] ( = d), and
BRi=((A,B)IA,B~Bil(n,q) and rank (A-B)=i}, HRi=~(A,B))A,BEHer(n,q) and rank (A-B)=i},
where O<i<n. Both (Bil(n,q), {BRilOdidn}) and (Her(n,q),(HR,(O<i<n)) are (P&Q)-polynomial association schemes of n-classes and, indeed, BT = (Bil(n, q), BR,) and HT=(Her(n, q), HR1) turn out to be distance-regular graphs of diameter n, respectively. Similarly, (Alt (n, q), { Rt 10 < i < dj ) and (Sym(n, q), { Ri I 0 < i < d)) are also (P&Q)-polynomial association schemes of d-classes; moreover, AT = (Alt (n, q), AR,) and ST=(Sym(n, q), SRI) turn out to be distance-regular graphs of diameter d and, surprisingly, they share the same intersection array, i.e.,
bi=q4i(q-l)(q”-2i-l)(q”-2i-q)/(q2-l)(q2-q), O<i<d-1 and
Ci = q 2i-2(q2i- l)/(q’- l), 1 <i<cl.
Recall that, for a distance-regular graph r of diameter d, bi = ) Ti+ 1 (x)nT, (y)l for Obidd-1 and ci=Iri_r(x)nT,(y)l for l<i<d, where x, y~V(r) are at distance i and Ti(x)={zlZEV(Z) and d(x,z)=iJ.
The above observations show that the distance-regular graphs defined over Alt(n, q) and Sym(n,q), respectively, are not characterized by their intersection arays, nor by the diagram geometries they belong to.
5. Alternating-forms graphs
Toward the goal of a geometric classification of the family of distance-regular graphs Alt(n, q), in this section, we shall provide more detailed analysis of the local structure of Alt(n, q). The notion of pseudo-alternating incidence structures is intro- duced in the hope that, in addition to its diagram, the geometry d =(,al,, &r, . . , ~2,) and its adjacency graph Alt(n,q) could be characterized in terms of it.
Let us start from the maximal cliques of the graph Alt(n,q). Since it is dis- tance-transitive, we are concerned only about those maximal cliques which contain the zero form.
250 T. Huang. M. Laurent
Theorem 5.1 (Hemmeter [7]). If%? is a maximal clique in Alt (n, q) which contains the
zero form, then either ufE K Rad( f) is contained in a hyperplane of V or &,, Rad( f)
is an (n - 3)-dimensional subspace of V.
Hence, up to isomorphism, there are two types of maximal cliques in Alt(n,q); cliques of the first type are of size q”-I, and the others are of size q3. If (fn-r, Un-l)E.dm-l, i.e., U,_ 1 s V is a subspace of dimension n- 1 and fn_ 1 EAlt(U,_ r), without loss of generality, we may assume thatf,_ 1 is the zero form.
The nth shadow a,(f,_,, U,_,) of (fn_t, U,_,) is defined to be ((fT V)(feAlt(n,q) and f 1 un_, =fn _ 1 }. An immediate consequence is the following corollary.
Corollary 5.2. For each (fnml, Un_l)~~n_l, its shadow ~,,(f,_~, U,_,)~Alt(n,q) is
a maximum clique of Alt(n, q) of size q”- ‘, and vice versa.
Remark. A similar result holds for _!Pq( U, V) and the distance-regular graphs H, (d, n), as mentioned in Section 4.
Before studying incidence structures related to Alt(n,q), it will be convenient to recall some notions about incidence structures (9,93,~) with the property that IBnB’(=O or p for distinct B,B’EGI. Let ~={B~B’IB,B’E~I are distinct and
BnB’ #o}, then 9’ S(Z) and elements of 9 will be called lines. A subset S 5 9’ is called
block-closed if 1 SnBl >p+ 1 for block BEG implies that BG S. Similarly, a subset S c 9 is called line-closed if 1 SnLl 2 2 for line LEE implies that L G S. A subset S s 9’ is called connected if any two points x, y of S are in a common block. A subset S c 9 is called a 2-subspace of ll if S is a smallest connected block-closed subset of 9 which contains two intersecting blocks. For any point y, I-r(y) is defined to be (z ( ZEP and y, z are in a common block}. For YGP, the set of common neighbors of Y is defined to be nueurl(Y).
Let I7 = (Alt(n, q), 69, E), where 9I is the set of all maximum cliques of the alternating forms graph Alt (n, q), i.e.,
~={~,(fn-l,Un-l)l(fn-l,
Un-l)-fn-lIC
( >
Ay;q)
If 9 is a flag of type {O,l,..., n-2} in the geometry d=(dcP,,dI ,..., &J and
(fn_2,
U,_2)E&c4n_zng, then the dual of Res(5) is a (q”-*,q+ l;q)-net satisfying (*), as mentioned in Section 4. Furthermore, any two intersecting blocks of 93 uniquely determine some(
fnp2,
U,-,)~d,- 2; it follows that [lo] the smallest connected block-closed subsets of Alt(n,q) which contain these two intersecting blocks is Res(F)), i.e., its dual is a (qn-*, q + 1; q)-net satisfying (*), where 8 is a flag of type {0,1,...,n-2},with(f,_~,U,-~)~BnA,_,.Morespecifically,ifB,B’aretwoblocksmeeting at a line A, then there exist Biggi (0~ i<q), B= Bj, B’= Bk for some
O<j,k<q such that noGiG, Bi = A. Let y be a point in B0 but not in A; then rI (y)nBi
(s, r; p)-nets and alternating forms graphs 251
CIJ
lQi<q r,(Y)nBi)U(r,(z)nBo)) f orms a second-type maximal clique of size q3 inAlt(n, q), where z is a point in Tl(y)nB1. Therefore, some features of the incidence structure Ii’=(Alt(n, q), a, E) can be summarized as follows:
(1) each line consists of q”- ’ points and each point lies on (q”- l)/(q- 1) blocks, (2) for any two distinct blocks B, B’E~?, IBnB’j =0 or q,
(3) any 2-subspace is a (qnm2, q + 1; q)-net satisfying (*),
(4) if B, B’E~? are two blocks meeting at a line A, and x is a point in B’ but not in A, then the common neighbors of elements of Au
jx}
form a clique of q3 points in Alt(n,q) (i.e., an analog of the dual of the Pasch axiom holds).The conditions for the following class of incidence structures are abstracted from the above observations; (Alt(n, q), 99, E) provides such an example with s = q”-’ and p = q. An incidence structure Il = (9”,9#, E) is called pseudo-alternating if the following conditions are satisfied:
(PAl) Each point lies on exactly (s,u~ - l)/(,~ - 1) blocks.
(PA2) For any two distinct blocks B and B’ in %?, (BnW =0 or ,u.
(PA3) If x,yEB,,nB1, then there exist Bi, 2<i<p, such that x, y~n,,~~~ Bi,
Ino,iG,BiI=~
and Uo<iqp Bi-
(x, y} consists of all common neighbors of x and y.(PA4) Any 2-subspace of Il is a (s, p + 1; p)-net satisfying (*) such that, for any two intersecting blocks B, B’ meeting at a line AE_Y and XEB’--A, the common neighbors of Au(x) form a clique.
Some consequences of the existence of such incidence structures are: (1) /J - 1 must be a divisor of s.r- 1.
(2) Each block consists of sp points, and each block can be partitioned into s lines. (3) Each line is contained in ,M+ 1 blocks of 99, and the intersection of those p+ 1 blocks is the line itself.
It seems worthwhile to state other properties of pseudo-alternating incidence structures as a formal proposition.
Proposition 5.3. (1) For each incident point-block pair (x, B), B- {x} can be partitioned into (su - l)/(c( - 1) subsets such that the union of(x) with each such subset is a line in 2’
(2) For each block B, the induced incidence structure II78 = (B, Y(B), E) is linear, where _Y(B)={B~B’IB’E.@ and BnB’#@}.
Proof. By (PA3), for each line A with XEA and A E B, in addition to B, there are exactly another p blocks which contain A. Hence, (1) follows from (PAl), and (2) follows from (PA2) and (1). 0
The next proposition treats the possible structures over 2-subspaces of pseudo- alternating incidence structures.
Proposition 5.4. Zf I72 =(X, B, E) is a 2-subspace of Il=(P, 9, E), then n is a prime power, II,/ z is a 2-attenuated space and .s=,tT1 for some integer n.
252 T. Huung, M. Laurent
Proof. Since IZ,/ N is a (s, r; 1)-net which satisfies the dual Pasch axiom, the proposi-
tion follows immediately from [15]. q
We conclude this paper by mentioning the following result without proof: If p 2 5, B~93 is a block and Ti(x)nB is either empty or line-closed in Z778=(B, _‘?((B),E) for xc.9 not in B, then ~1 is a prime power and Z778=(B, 2((B), E) is isomorphic to the affine space AG(n - 1, q) for some integer
n.
Acknowledgment
Both authors thank Professors M. Deza, Koh-Wei Lih and the Institute of Mathe- matics, Academia Sinica, Taipei, for providing a multitude of stimulating contacts during the preparation of this paper.
References
[l] E. Bannai and T. Ito, Algebraic Combinatorics I. Association Schemes, Benjamin/Cummings Lecture Note Series in Math. (Springer, Berlin, 1984).
[2] F. Buekenhout, Diagrams for geometries and groups, J. Combin. Theory Ser. A 27 (1979) 121-151. [3] H. Cuypers, Two remarks on Huang’s characterization of the bilinear forms graphs, preprint. [4] P. Delsarte, Association schemes and r-designs in regular semilattices, J. Combin. Theory. Ser. A 20
(1976) 230-243.
[S] M. Deza and M. Laurent, Bouquets of matroids, d-injection geometries and diagrams, J. Geom. 29 (1987) 12-35.
[6] H. Hanani, On transversal designs, in: Combinatorics, Math. Centre Tracts 55 (Mathematisch Centrum, Amsterdam) 42-52.
[7] J. Hemmeter, Distance regular graphs and halved graphs, European J. Combin. 7 (1986) 119-129. [S] T. Huang. A characterization of the association schemes of bilinear forms, European I. Combin.
8 (1987) 1599173.
[9] T. Huang, An analogue of the Erdbs-Ko-Rado theorem for the distance regular graphs of bilinear forms, Discrete Math. 64 (1987) 191-198.
[lo] T. Huang and Y. Pan, A diagram geometry for the association schemes of alternating bilinear forms, Proc. of NSC, Part A 12 (Rep. of China, 1988) 44-49.
[11] D. Jungnickel, Latin squares, their geometries and their groups. A survey (1988) (A revised and updated version of Lateinis the Quadrate, ihre Geometrien und ihre Gropen, Jahresber, DMV 86, 69-108).
[12] G.L. Mullen, Polynomial representation of complete sets of mutually orthogonal frequency squares of prime power order, Discrete Math. 69 (1988) 79984.
[13] D.J. RayyChaudhuri and N. Singhi, On existence and number of orthogonal arrays, J. Combin. Theory Ser. A 47 (1988) 28836.
[14] A.P. Sprague, Incidence structures whose planes are nets. European J. Combin. 2 (1981) 193-204. [15] J.A. Thas and F. DeClerck, Partial geometries satisfying the axiom of Pasch, Simon Stevin 51 (1977)
123-137.
