Algebraic codes and geometry of some classical generalized polygons

Transcription

1 isibang/ms/2013/20 August 20, statmath/eprints Algebraic codes and geometry of some classical generalized polygons N.S.Narasimha Sastry Indian Statistical Institute, Bangalore Centre 8th Mile Mysore Road, Bangalore, India

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3 Algebraic codes and geometry of some classical generalized polygons N.S.Narasimha Sastry Division of Theoretical Statistics and Mathematics Indian Statistical Institute 8 th Mile, Mysore Road, R.V. College Post Bangalore India Abstract Some results about the geometry of, and the q- ary codes associated with, finite generalized polygons W (q), q = 2 n ; H(q), q = 3 n ; and O(q), q = 2 2m+1, are surveyed and a few simple observations and related questions are mentioned (in italics). Using the polarity in the cases W (q), q = 2 2m+1, and H(q), q = 3 2m+1, we define a nondegenerate symmetric form on, and a polarity of, the q ary code associated with these geometries which is stabilized by the centralizer of the polarity in the automorphism group of the geometry. 1 Introduction (i) In this article, we consider the q - ary codes, q = p n, p a prime and n 1, related to the finite generalized polygons X = (P, L): mostly the regular (q, q)- generalized quadrangle (GQ, for short) W (q), q = 2 n and a few remarks about the regular (q, q)-generalized hexagon (GH, for short) H (q), q = 3 n, and the (q, q 2 ) - generalized octagon (GO, for short) O (q), q = 2 2m+1. The automorphism groups of these geometries contain subgroups G isomorphic to P SP (4, 2 n ) if X = W (2 n ), to G 2 (3 n ) if X = H(3 n ) and to the Ree group G = 2 F 4 (2 2m+1 ) if X = O(2 2m+1 ). In each case, G acts transitively on the point - set P, the line-set L and the set F ={(x, l) P L : x l} of all flags of X. Recall that a generalized n gon is a bipartite graph (undirected, with no loops and multiple edges) of diameter n and girth 2n. To all unexplained terminology and results, see the standard references [25], [37] and [50]. These codes are of interest because of their close connection to the rich geometry defining these groups. See [3],[5],[6] for an application to study ovoids in the projective 3- space over F 2 n. In this note, 1

4 we record a few simple observations and put together some related questions. These codes are also of interest because they are interesting submodules of some permutation kh- modules with large indecomposable sections, where k is a field and H is a finite group of Lie type defined over a field of the same characteristic as k: the k - row span of the incidence matrix of a Lie incidence system of H whose point-set and the line-set are objects of two given distinct types of the Lie geometry for H with respect to a H- invariant incidence relation; more generally the image and the kernel of a kh- homomorphism between permutation kh- modules. (ii) Let k be an algebraically closed field of characteristic p. For M {P, L}, let k M denote the kg- module of all functions from M to k with the action of G defined by (g f)(x) = f(g 1 x) for g G, f k M and x M. For A M, we denote by 1 A k M the function taking x M to one if x A and zero otherwise. For a collection of subsets of M, by abuse notation, we denote by the k- subspace of k M generated by {1 A : A }. We write C for L k P. Since ( M, p) = 1, the map c from k M to k taking f k M to its content Σ m M f(m) has a splitting taking λ k to λ1 A k M. So, we have a kg module decomposition k M = k1 M Y, where Y = ker c. For any subset A of k P, we write A Y as A 0. Since the number of lines in X containing any given point of X is 1(mod p), 1 P C and C = k1 P C 0. Being a permutation module k P, and so Y, are self-dual. For any subset A of P, we denote by A the orthogonal of A with respect to the inner product f on k P for which the characteristic functions of singleton subsets of P form an orthonormal basis. (iii) In the generalized n gons X above, each pair of points distance n apart is contained in, and determines, a unique (1, t)- subgon of X ( where t = q 2 if X = O(q) and t = q if X = W (q) or H(q)). There is a natural partition A B of the point-set P 1 of a (1, q) - subgon of X, where A and B are the equivalence classes in P 1 with respect to the equivalence relation : x y if, and only if, d(x, y) = 0 or 4 (mod n). The incidence system (A, B), with a A incident with b B if a and b are collinear in X, is a complete bipartite graph with A = B = q + 1 and A and B are lines, not of W (q), forming a dual grid described in ( 3.B.iv) if X = W (q); A and B 2

5 are Desarguesian Projective planes of order q if X = H(q) (see [41],.6, for this in terms of the ideal line condition ) and the GQ W (q) if X = O(q). Further, each word of minimum weight in C appears as a nonzero scalar multiple of 1 A 1 B, where A B is a partition as above of a point set of a (1, t)-subgon of X (see, for example, [42]). Our first question asks for the kg-module structures of : (i) the subspace of k p generated by {1 A }, where the sets A P are as above; and (ii) the subcode of C generated by its words of minimum weight. This is a proper subcode of C if X = W (q), q > 2 ( 3.D. first para of i). Finally, I dedicate this to the memory of saintly and maternal figures in my life Kanakammagaru and Venkatalakshammagaru. 2 The GQ W (q), q = 2 n (A) Geometry of W (q): A duality of P G(3, q) is an inclusion reversing bijection of the set of all subspaces of P G(3, q). It interchanges the set P of points and the set Π of planes of P G(3, q), and fixes the set L 0 of all lines of P G(3, q). A duality of order 2 is a polarity. Given a polarity τ of P G(3, q), a subspace A of P G(3, q) is said to be absolute (relative to τ) if it is incident with τ (A). We say that τ is a null polarity if each point (and so, each plane) is absolute. The set of absolute lines of P G(3, q) with respect to a null polarity is called a general linear complex. We denote by L the set of all general linear complexes in P G(3, q). A polarity of P G(3, q) is induced by a nondegenerate sesquilinear form f on F 4 q. It is a null polarity if f (x, x) = 0 for each x F 4 q ([25],p.42-43). We note that the natural action of P GL(4, q) on the set of all null polarities on P G(3, q), and so on L, is transitive. Further, the stabilizer of a member of L is isomorphic to P SP (4, q). If τ is a null polarity of P G(3, q) and L L is the set of absolute lines with respect to τ, then the incidence system X = (P, L) is a (q, q) - GQ (see [37], 3.1.1), denoted by W L (q) (or W (q) if L in question is clear). Recall that an (s, t) - generalized quadrangle is an incidence system X 1 = (P 1, L 1 ), where P 1 is the point set and L 1 is the line set with any two points incident with at most one line, s + 1 points incident with each line, t + 1 lines incident with each point and, for each pair of nonincident point - line pair (p, l), there is exactly one line containing p and sharing a point with l. Then, P = (s + 1) (st + 1) and L = (t + 1) (st + 1). In the GQ W (q) = (P, L) defined by a polarity of P G(3, q) as above, for x P, π x = x l L l = τ (x) Π. A duality of an incidence system X 1 = (P 1, L 1 ) is an incidence preserving 3

6 bijection of P 1 L 1 which interchanges P 1 and L 1, thus defining an isomorphism between X 1 and its dual incidence system X 1 = (L 1, {L p : p P }), where L p = {l : p l L}. A polarity of X 1 is a duality of order 2. For a polarity τ, an element of P 1 L 1 incident with its image under τ is said to be absolute. For each n, W (2 n ) admits a duality (see [37], 3.2.1). It admits a polarity if, and only if, n 1 is odd (Tits, [49]).To understand this geometry, Klein correspondence (see [30], Chap.15) and the fact that the tangent lines of a nondegenerate quadric in P G (2n, q) for even q and n 1 meet at a point (called the nucleus of the quadric) ([31], Corollary 2, p.10) are very useful. By Klein correspondence, we have a useful (α) isomorphism from the dual of W (q) (and so, from W (q)) to the (q, q)- GQ Q (4, q) of points and lines contained in a nondegenerate quadric E in P G(4, q); and (β) an embedding of Q (4, q) (and so, of W (q)) as a subgq of the (q, q 2 ) GQ Q (5, q) of points and lines of an elliptic quadric Ẽ in P G(5, q) by realizing E as Ẽ π for a hyperplane π of P G (5, q). We note that each line of Q (5, q) is either a line of Q (4, q) or contains exactly one point of Q (4, q). We briefly explain. Under Klein correspondence, lines of P G (3, q) are mapped bijectively on to the points of a hyperbolic quadric K of P G (5, q); for L L, members of L to the points of a nondegenerate quadric E = π K of π, where π is a hyperplane in P G (5, q); and elements of L through a point of P G(3, q) to points of a line contained in K π. This gives an isomorphism from the dual of W L (q) to the GQ Q (4, q) mentioned in (α). For (β), we choose an elliptic quadric Ẽ in P G (5, q) with Ẽ K = E. To briefly describe the polarity of W (q), we fix an embedding of P G(3, q) as a subspace A of P G (5, q). Ensuring that the nucleus n of Q is not in A, we project from n the points and lines of E to P G (3, q) yielding points and lines of a GQ isomorphic to W (q) embedded in P G (3, q). Applying a collineation of P G(3, q), we obtain a duality of W (q). If ϕ is a polarity of W (q), then ϕ ψ 1 is a collineation δ of P G(3, q). Requiring that δ ψ be an involution, we get a polarity of W (q). (B) Ovoids and spreads in P G(3, q) and in W (q): (i) An ovoid of P G(3, q), q > 2, is a set of q points no three collinear. An ovoid of P G(3, 2) is a set of five points, no four coplanar ([25]). By definition, an elliptic quadric in P G(3, q) is an ovoid of P G(3, q). For q odd (a case not discussed further here), these are the only ovoids in P G(3, q) (independently 4

7 due to Barlotti and Panella, see [10] and [36]). We denote by O (respectively, E) the set of all ovoids (respectively, all elliptic ovoids) of P G(3, q). An ovoid of W (q) is a set of its q pairwise noncollinear points. An ovoid of W (q) is also an ovoid of P G(3, q) ([37],1.8.2). By a fundamental Theorem due to Segre (see [33], 25.10, p.127), if q is even, the set L of all tangents to an element θ of O is a general linear complex and θ is an ovoid of the GQ W L (q). So, classification of ovoids of P G(3, q) and ovoids of W (q) are equivalent. We denote by O L the set of all ovoids in P G(3, q) with L as the set of its tangent lines. Write E O L as E L. Clearly, the sets {O L } L L of ovoids of P G(3, q) partition O. A collection of q lines of P G(3, q) partitioning P is called a spread of P G(3, q). It is called a symplectic spread of P G(3, q) or a spread of W L (q) if all these lines belong to a general linear complex L. We recall the important concept of a regular spread. Given any three pair-wise skew-lines l 1, l 2, l 3 in P G(3, q), there is a unique transversal to l 1, l 2, l 3 (that is, a line of P G(3, q) meeting each l i ) through any point on any one of the lines l i. If R is the set of the q + 1 transversals of l 1, l 2, l 3, then a transversal to any three members of R is a transversal to all members of R. The set R of all transversals of members of R is called the regulus in P G(3, q) determined by l 1, l 2, l 3. A spread S in P G(3, q), q > 2, is said to be regular if it contains the regulus in P G(3, q) determined by each triple of its members ([25], 5.1); equivalently [40], if, for some l 1, l 2 S, l 1 l 2, S contains the regulus determined by l 1, l 2, l 3 for each l 3 S\{l 1, l 2 }. Any regular spread is symplectic. An ovoid in W (q) is elliptic if, and only if, its image under a duality of W (q) is a regular spread of W (q). Under Klein correspondence, a regular spread S of P G(3, q) is mapped on to an elliptic quadric K l O (3, q) in the 3-subspace l of P G (5, q), where K is the Klein quadric and l is a line of P G (5, q) disjoint from K. Each of the q + 1 hyperplanes of P G(5, q) containing l intersects K in a nondegenerate quadric isomorphic to Q (4, q). Under Klein correspondence, these are precisely the hyperplane sections of K corresponding to the q + 1 members of L containing S. (ii) Classical ovoids: (a) Elliptic ovoids: An elliptic quadric in P G(3, q) is an ovoid of W L (q), L L, if, and only if, its defining quadratic form polarizes to a nondegenerate symplectic bilinear form on P G(3, q) such that L is the set of its isotropic lines. Each elliptic ovoid of W L (q) can be viewed in the following useful ways: 5

8 (α) as (exactly) one of the q + 1 orbits for the semi-regular action on P of a cyclic subgroup of Aut(W L (q)) of order q (see [26], (B.vi.a)) (β) as the set E x E of all points of Q(4, q) (isomorphic to W (q), see (A.β) and the notation there) collinear to a given point x Ẽ of the (q, q2 ) - GQ Q (5, q) containing Q(4, q) as a hyperplane intersection, but x not a point of Q(4, q). An elliptic quadric θ of Q(4, q) is E x for some x Ẽ\E if, and only if, x belongs to the secant line θ of Ẽ (Metz, see [24]). Thus, there is a 2 to 1 map from Ẽ\E on to the set of all elliptic ovoids in Q(4, q) and E L = Ẽ\E /2 = q2 (q 2 1) /2. By Klein correspondence, any two elliptic ovoids of Q(4, q) intersect either in a point or in a conic of E. (γ) If l is a line of Q (5, q) meeting E at a point p, then the set E l = {E x : p x l} of q elliptic ovoids of Q (4, q), together with the lines of Q (4, q) incident with p, define a partition of P \{p}, a fact used below and in the last para of 3.D.i. Further, E l E L is a maximal in E L with respect to pairwise intersection at p. Also, for lines l and m of Ẽ meeting E at a point, E l = E m if, and only if l, m and the nucleus of E in Ẽ are coplanar. Let y Ẽ\(E l). If y is collinear to p Q (4, q), then either y θ for some θ E l and so E y = θ ; or θ E y is a conic in Q (4, q) containing p for each θ E l. If y is not collinear to p, then p / E y, q 2 members θ of E l intersect E y in mutually disjoint conics, remaining two members of E l intersect E y in at distinct points and the remaining q +1 points of E y are the points of Q(4, q) collinear to both y and p. Thus, given an elliptic ovoid θ (= E x ) in Q(4, q) and p θ, θ is in a unique set C θ,p (namely, E px ) of q elliptic ovoids of Q (4, q) which is maximal with respect to pairwise intersection at p. Further, {C θ,p \{θ}} p θ is the set (having (q 1) (q 2 + 1) elements) of all elliptic ovoids of Q(4, q) which intersect θ at a point. Let θ 1 and θ 2 be elliptic ovoids of Q(4, q). If θ 1 θ 2 = {p}, then C θ1,p = C θ2,p and each of the q 2 members of C θ1,p\{θ 1, θ 2 } intersects both θ 1 and θ 2 at p. Any elliptic ovoid θ of Q (4, q) intersecting θ 1 at a point x p belongs to C θ1,x\{θ 1 }. Restricting the partition of P \{x} described in (γ) to θ 2, we see that there is a unique member θ of C θ1,x\{θ 1 } intersecting θ 2 at a point different from p (and each of the remaining q 2 members of C θ1,x\{θ, θ 1 } intersect θ 2 in a conic and each line m of Q (4, q) through x intersects θ 2 at a point). Thus, there are q 2 + (q 2) elliptic ovoids of Q(4, q) intersecting each 6

9 of θ 1 and θ 2 at only one point. If θ 1 θ 2 is a conic, then, by the uniqueness stated above, C θ1,p and C θ2,p are disjoint. Further, if x θ 1 \θ 2, again by (γ), there are exactly two members of C θ1,x intersecting (θ 1 at x and) θ 2 at distinct points of θ 2 \θ 1. Thus, there are 2 (q 2 q) elliptic ovoids of Q(4, q) intersecting each θ i, i {1, 2}, at a point. (b)tits ovoids: If q = 2 2m+1, m 1, the set of absolute points (respectively, all absolute lines) of a polarity of W (q) is an ovoid (respectively, spread) of W (q) (Payne, see[37],1.8.2), called a Tits ovoid (respectively, Lüneburg spread, see [33]). Since composition of two polarities of W (q) is an automorphism of W (q), Aut (W (q)) P ΓSP (4, q) acts transitively on the set T L of all Tits ovoids of W (q) = W L (q). Its action on the set E L is also transitive. Tits ovoids and elliptic ovoids in P G(3, q) are distinct for q 8 because their stabilizers in P SP (4, q), being respectively isomorphic to 2 B 2 (q) ([49]) and P SL 2 (q 2 ) 2 ([30]), are nonisomorphic. Elliptic ovoids (which exist for all q) and Tits ovoids (which exist only for all odd powers of 2) are the only known ovoids of P G(3, q). They are collectively called classical ovoids. An ovoid of W (q) admits a transitive group of automorphisms if, and only if, it is classical ([3],Theorem 1, see also [23]). For q {2, 4, 16}, each ovoid in P G(3, q) is elliptic. For q {8, 32}, each ovoid in P G(3, q) is either elliptic or of Tits type (see [12]). A recent remarkable development in this direction due to Pentilla is that, if q is a power of 4, then any ovoid in W (q) is elliptic ([38]). I thank Pentilla for this communication. (iii) In view of the connection of ovoids of P G(3, q) to several other combinatorial structures (for example, inversive planes, see [33], 25.6, p.126; translation planes [48]; maximal arcs [48]; unitals [16], [35]; association schemes; group divisible designs [15]; semi-biplanes [4]; Tits (q, q 2 )- generalized quadrangles [37], etc), study of various structures on O and their embeddings in P G(3, q) may be fruitful. Questions like classification of ovoids, intersection pattern of members of O and structure of association schemes on O and on its suitable subsets; partitions of the point set P of P G(3, q) by ovoids or by q 1 hyperbolic quadrics and 2 lines, etc. could be of interest. Identifying more differences between elliptic and Tits ovoids (like, for example, their secant plane sections ([12]), existence of special tangent lines ([25], , p.53), the subcodes of k P generated by their characteristic functions [43], etc) may yield pointers towards the nonexistence (conjecturally) of nonclassical 7

10 ovoids. (iv) Dual grids in W (q): We denote by H the set of all hyperbolic quadrics in P G(3, q). For L L, let H L denote the set of the members H of H such that each of the 2 (q + 1) lines in H is in L. For each H H L, the duals in W L (q) of the two parallel classes of lines in H (containing q + 1 elements each) are the lines m, m of P G(3, q) not in L. Here, for any subset A of P, A P denotes the set of elements of P collinear in W L (q) to each element of A. The subset m m of P is called a dual grid of W (q). It has the following equivalent descriptions: (α) the set of all points of a (1, q) - subgq of W L (q); (β) m = x y and m = z tr{x,y} z for any two distinct points x, y of m; and (γ) if θ is any ovoid of W L (q), then one of m, m is a secant to θ meeting θ at, say, {x, y} and the other is x y, the intersection of the tangent planes to θ at x and y. (v) Ovoidal partitions of P : For a dual grid m m, by (iv.γ), m and m meet each ovoid of W L (q) in 0 or 2 points. Since q is even, P cannot be partitioned by ovoids of W L (q). (a) Existence of ovoidal partitions: Subgroups of A, A {P SL(4, q), G}, P SP (4, q) G < P SL(4, q), of order q form a conjugacy class in A and the centralizer in P SL(4, q) of such a subgroup T is a Singer cycle T = T K, where K is a subgroup of order q + 1. So, T and K act on P as well as on the set of lines of P G(3, q) semi-regularly. The orbits θ 0,, θ q of T in P are elliptic quadrics in P G(3, q), defining a partition of P ([26], Theorem 3, p.1167). Any two such subgroups of P SL(4, q) have at most one orbit in common ([1], Lemma 4.1). If T Aut (W L (q)) for some L L, then only one of the θ i s, say θ 0, is an ovoid of W L (q). One of the line orbits of T is a regular spread S and is the set of all lines of P G(3, q) tangent to each θ i, i {0,, q}. The orbits of K in P are the elements of S. (b) Let {θ i } 0 i q be a set of ovoids of P G(3, q) partitioning P and L i L denote the set of tangent lines to θ i.then, a line l of P G(3, q) is tangent to each θ i or it is tangent to a unique ovoid θ l, secant to q/2 of the other ovoids and passant to q/2 remaining ovoids. Klein correspondence implies that there is a regular spread S of lines of P G(3, q) such that L i L j = S for i j and {L 0,, L q } are the only general linear complexes in P G(3, q) containing S (see [17], 2 and Lemmas 2.4 and 2.5). On the other hand, any regular spread S of P G (3, q) is contained in precisely q + 1 general linear complexes and the cyclic group K of collineations of P G (3, q) fixing each 8

11 member of S acts regularly on these general linear complexes. Further, if θ is an ovoid of P G(3, q) such that each member of S is tangent to θ, then {g (θ) : g K} is a partition of P (see [17], Lemma 2.5 and Theorem 3.1). In particular, there is a partition of P by Tits ovoids (see [22], Theorem 7). It is not clear if all partitions of P appear in this way (equivalently, whether a cyclic group of SP (4, q) of order q + 1 acts regularly on the ovoids of a partition). Existence of partitions of P by ovoids of P G(3, q) of different types is not known (see [22], 2.3). (vi) We present a few facts about O. For θ O, let θ denote the set of all planes tangent to θ and L (θ) L denote the set of all lines tangent to θ. Proposition 1 Let θ 1, θ 2 O. Then, (α) (Butler [18],Theorem 12) θ 1 θ q2 + 1; (β) (Bruen and Hirschfeld [13],Theorem 5.1) θ 1 θ 2 = θ 1 θ 2 ; and (γ) (Butler [18], Lemma 2.2) if L(θ 1 ) L(θ 2 ), then L(θ 1 ) L(θ 2 ) is either a regular spread in P G(3, q) or a set of q 2 + q + 1 lines meeting l (including l itself) or a set of (q + 1) 2 lines of a common (1, q) -sub GQ (equivalently, lines of a common dual grid). Pentilla has found two Tits ovoids of P G(3, 8) with 33 points in common (private communication). However, the bound in (α) seems to be excessive for general q. Possible intersections of classical ovoids with different sets of tangent lines in P G(3, q) is not known. Analysis in ([27], III.C) using the classification of pencils of quadrics in [14] shows that any two elliptic ovoids of P G(3, q) meet in at most 2 (q + 1) points. In (β), the points of tangency in θ 1 and θ 2 of an element of θ1 θ2 need not be in θ 1 θ 2 (see [7] for a discussion). It would be interesting to see if there is a natural bijection between θ 1 θ 2 and θ1 θ2. More generally, the number of points common to two nonsingular quadrics of the same type in a finite projective space of odd dimension is equal to the number of their common tangent hyperplanes. This is not true for finite projective spaces of even dimension, even for projective planes (there are disjoint ovals with common tangent lines (see [13], p.218 and [7], Lemma 2.1). Though there exist ovals in a projective plane intersecting at a point with distinct tangent lines at the point of intersection, I do not know if a pair of ovoids in P G(3, q) intersecting at a point with distinct tangent planes at the point of intersection can exist. 9

12 To see (γ), we note that the Klein correspondence τ maps L(θ 1 ) L(θ 2 ) on to the intersection of a projective 3-subspace B = L(θ 1 ) L(θ 2 ) of the projective 4 space L(θ 1 ) with the nondegnerate quadric L(θ 2 ). The possibilities for the later are : an elliptic quadric in B, a hyperbolic quadric in B and a quadratic cone ([30], chap.15). They correspond under τ to the possibilities in (γ). (Vii) Intersection of classical ovoids in W L (q) W (q), q = 2 2m+1 8, L L: We first describe the subsets of a Tits ovoid θ of W L (q), each of which appears as the intersection of θ with some Tits ovoid θ of W L (q). If H 2 B 2 (q) is the stabilizer of θ in P SL(4, q), then H < Aut (W L (q)) and H = q 2 (q 2 + 1) (q + 1). The stabilizer in H of a point x of θ is P K, where P Syl 2 (H), K C q 1 and the action of P on θ \ {x} is regular (see [34], p. 90). Union of {x} and an orbit of the centre of P in θ is an oval and is called a pseudo-circle in θ. Union of two distinct pseudo-circles in θ through x is called a figure of eight in θ. H contains a unique conjugacy class A + (respectively, A ) of cyclic subgroups of orders q + r + 1 (respectively, q r + 1), where r 2 = 2q. The normalizer in H of each of these groups is maximal in H and is of order 4 (q + r + 1) (respectively, 4(q r + 1)). Each member of A + A is self-centralizing in H and its centralizer in P SL (4, q) (respectively, in Aut (W L (q))) is a cyclic subgroup T of order P (respectively, a subgroup T of the Singer cycle T of order q 2 + 1) (). see [34], Theorem 3.10, p.190). As noted in (ii.α), the T orbits in P are elliptic quadrics and only one of them is an ovoid of W L (q). Since this accounts for H /4 (q + r + 1)+ H /4 (q r + 1) = q 2 (q 2 1)/2 elliptic ovoids in W (q) ( subgroups of order q of Aut (W L (q))), the map T T H is a bijection from the set of subgroups of Aut (W L (q)) of order q and A + A. Thus, H has two orbits in the set E L of all elliptic ovoids of W L (q) (and two orbits on the set of regular spreads in L, in view of the polarity of W L (q) centralized by H) with 4 (q + r + 1) and 4 (q r + 1) elements. An A- orbit in θ is called a cap or a cup according as A A + or A. Each A- orbit in θ is the intersection of θ with the T - orbit E in P containing it. For use in ( 3.C), we note that 1 E = Σ x T 1 x(θ) ([43], Lemma 14). Theorem 2 Let L be a general linear complex in P G(3, q) and θ 1, θ 2 O L be distinct. Then, the following hold: (i) (Glynn [28]) θ 1 θ 2 q(q 1)/2. (ii) (Butler [17]) θ 1 θ 2 is odd. Further, θ 1 θ 2 = 1 (mod 4) if θ 1 is 10

13 an elliptic ovoid. (iii) (Glynn [29]) If θ 1 and θ 2 are both elliptic, then θ 1 θ 2 is either a point or a conic (that is, the intersection of θ i with a nontangent plane). (iv) (Bagchi and Sastry [5]) If θ 1 is elliptic and θ 2 is a Tits ovoid, then θ 1 θ 2 is either a cup or a cap. (v) (Bagchi and Sastry [5]) If θ 1 and θ 2 are both Tits ovoids, then θ 1 θ 2 is one of the following: a point, a pseudo-circle, a figure of eight, a cup or a cap. (See also [8]; [50], p.341 for (iv) and (v)).thus any two ovoids (dually, any two spreads) of W (q) have at least one element in common. If an ovoid of W (q) intersects each elliptic ovoid of W (q) in a point or a conic (respectively, a cup or a cap), is it necessarily an elliptic ovoid (respectively, Tits ovoid) of W (q)? Other possibilities for intersection of ovoids exist if the sets of tangent lines are different. Example: Let l m be a dual grid in the GQ W (q) = (P, L).The group Aut (W (q)) contains a cyclic subgroup L of order q 1 which fixes each point of l m and acts regularly on the (q + 1) 2 sets xy \ {x, y}, where x l, y m and the line xy L. Fix x 0 l, y 0 m and an ovoid θ of P G(3, q) containing x 0 and y 0. Then, aθ is an ovoid of P G(3, q) for each a L and the q 1 ovoids {aθ : a L} intersect pairwise at {x 0, y 0 }. (C) The graph Γ O : Let Γ O denote the graph with vertex set O and θ 1, θ 2 O defined to be adjacent if θ 1 θ 2 = 1. For A O, we denote by Γ A the subgraph of Γ O induced on A. Properties like connectedness, regularity, diameter etc. of Γ A for A {O, E, J, E L, J L for L L} would be interesting. We make some remarks about their cliques. Lemma 3 If q 4, a clique in Γ O has at most q + 1 vertices. Proof. If there were q + 2 pairwise adjacent vertices in Γ O, then their union would have at least (q + 2) (q 2 + 1) ( ) q+2 2 points. This exceeds P by q(q 3), 2 a positive number if q 4. W (2) has 6 ovoids (all elliptic) pairwise adjacent. Let x P, π be a plane in P G(3, q) containing x and O x,π O denote the set of all ovoids of P G(3, q) containing x with π as their common tangent plane at x. Each 11

14 clique in Γ Ox,π of size q (in particular, the clique E l in Γ EL described in (B.ii.a.γ) is maximal in Γ O. We now show that there is an elementary abelian subgroup of P SL(4, q) of order q acting semi-regularly on O x,π whose orbits thus partition O x,π by maximal cliques of size q. Proposition 4 Let x, π and O x,π be as above and L L. Assume that L contains each line in π incident with x. Let G < Aut(W L (q)) with G SP (4, q) and M be the subgroup of G consisting of all elements of G fixing each point of π. Then, the following hold: (i) M is isomorphic to the additive group of F q. (ii) If m M fixes an element of P \ π, then m is trivial. (iii) M stabilizes each line l π of P G(3, q) containing x and acts regularly on l \ {x}. (iv) For θ O x,π and m M, m (θ) O x,π and {m (θ)} m M is a maximal clique in Γ O. Proof. With an appropriate choice of the symplectic basis for F 4 q for the standard representation of SP (4, q) on F 4 q, M is seen to be conjugate to {I 4 + λe 1,4 : λ F q }, where E 1,4 is the 4 4- matrix over F q whose (1, 4) th - entry is 1 and the rest are zero. So, (i) follows. If m M fixes z P \ π and l is a line incident with x, then m fixes l π. Since order of m is even, m fixes a third point of l not in {z, l π} and so is identity on l. Thus m is identity on each line through z and so on P, proving (ii). If l is as in (iii), then l z = π and, so l = ( l ) is fixed by each m M. Here, the perpendicularity is in W L (q). Now, (ii) completes the proof of (iii). For θ and m as in (iv), π is tangent to m (θ) at x. So, m (θ) O x,π. Each of the q 2 lines l π incident with x is secant to m (θ) and, by (iii), l m (θ) l m (θ) for distinct m, m M. So, the ovoids {m (θ)} m M are distinct and pairwise intersecting at {x}. Maximality of the clique follows because the complement of their union is π\{x}. Proposition 5 Let L L and q > 2. (i) (Hubaut, Metz [32]) Γ EL is a strongly regular graph with parameters v = ( ) q 2 2, k = (q 1) (q 2 + 1), λ = (q 1) (q + 2) = q 2 + (q 2) and µ = 2 (q 2 q). 12

15 (ii)any clique in Γ EL containing three members intersecting mutually at a common point is a subset of a maximal clique of the type E l described in (B.ii.a.γ). (iii) (Van Maldeghem and Sastry) Let A be a clique in Γ EL whose members meet mutually at distinct points. Then, A has at most 6 elements. Any clique of three elements can be extended to a clique of size 5 if q = 4 and to a clique of size 6 if q > 4. Proof. (i) is well-known. (ii) follows from the partition described in (B.ii.a.γ). For (iii), we use the set up outlined in the last two paragraphs of (B.ii.a.γ). It also contains a proof of (i). Let A be a clique in Γ EL with at least 3 elements θ 1, θ 2, θ 3. Let θi Ẽ = {a i, b i } Ẽ\E, i = 1, 2, 3. Then, a i b i and θ i = E ai = E bi. With proper indexing, we can take {a 1, a 2, a 3 } and {b 1, b 2, b 3 } to be triads in Q (5, q) and their traces are conics C and D, respectively. If θ 4 A and θ 4 θ i for i = 1, 2, 3, then θ 4 = E a4 = E b4 for non collinear points a 4, b 4 in Ẽ\E and, each of them is collinear to a i or b i (and not both) for i = 1, 2, 3. Let b 4 a 1, say. Then, as a 1 b 2, b 4 b 2. So, b 4 a 2. Similarly, b 4 a 3 and b 4 C. If A has a fifth element θ 5 = E a5 = E b5, then C contains an element b 5 collinear to a 4. Since the plane π containing C is not contained in a 4 (since b 4 a 4 ) and a 4 π is a line, there are at most two possibilities for b 5 and (iii) follows. 3 The code C Let X = (P, L) be the GQ W (q) and the notation be as in 1.i. (A) Socle of k P and of C: Since q and the characteristic of k are even and since the number of lines as well as the number of planes of W (q) incident with each x P is odd, 1 πx C and 1 P = Σ x P 1 πx Π C. We note that Π 0 is a simple kg - module; in fact, the Steinberg module for G. If θ is an ovoid of W (q), then the characteristic functions of its tangent planes {π x : x θ} form a basis for Π ([3], Lemma 6, p.144): in fact, for a secant plane π of θ, each p P is on q + 1, 1, 2 or 0 tangent planes to θ at points of the oval π θ according as p is the nucleus n of π θ; n p π; p / π and p lies on, or does not lie on, the tangent planes considered. So, 1 π = Σ x θ π 1 πx. For any subset A of θ, the restriction of Σ x A λ x 1 πx to θ is Σ x A λ x x. This is zero if, and only if, λ x = 0 for each x A. Thus, dimension of Π is q Since 1 P Π, Π 0 is of codimension one in Π. 13

16 Proposition 6 (i) Socle (k P ) = Π k1 P Π 0 k P /rad (k P ) and is of dimension q (ii) Socle (Y ) = Π 0 Y/rad (Y ). In particular, Y is indecomposable. Proof. Let S be a Sylow 2- subgroup of G. Then, S fixes a flag (p, l) in W (q) and so, also π p. Further, it acts transitively on the sets l \ {p}, π p \ l and P \ π p. So, 1 {p}, 1 l,1 πp and 1 P generate the space of S- fixed points in k P. Hence, k1 P and 1 P 1 πp : p P = Π 0 are the only simple kgsubmodules of k P and socle of k P is Π. From the preceeding remarks and the self-duality of k P and Y, rest follows. In particular, socle of C is Π, that of C is Π 0 and that of any kg submodule of k P is either k1 P, Π 0 or Π. Given a duality of δ of W (q), the map from P to C taking x P to 1 δ(x) defines a linear map δ from k P onto C with kernel C. Further, for l L, δ(1 l ) = 1 πδ(l) Π and δ is a bijection between P and L. So, C k P /C and C/C C Π. For the decomposition of the permutation module k P = k1 Y (as in 1.A) for the permutation action of a finite group G with a split (B, N)- pair defined over a field of the same characteristic p as that of k on the cosets of a maximal parabolic subgroup, Y and Y/rad(Y ) are simple kg modules. For details, see.11 and Remark 11.4 of ([2]). (B) Structure of C: The dimension of C is n [(1 + 17) 2n + (1 17) 2n ] = Σ I N 4 I, (*) where N is the collection of all subsets of Z/2nZ which do not have consecutive elements ([43], Theorem 1, p.485; for the last equality, see [21], p.34-35). This formula is obtained by identifying the composition factors appearing in each socle layer of C 0. Note that (*) is not a rational function of q. The lattice of submodules of C is also known ([43], p.491). As each kg- composition factor of C 0 appears precisely once, each submodule of C 0 is determined by the isomorphism type of its quotient by its radical. Further, the number of submodules of C 0 is finite and is independent of k. The composition factors of C 0 and of k P are the same ([46], Theorem 1, 238). Canonical generating sets for the socle layers of C 2 = 1 l : l L k P, considered as a F 2 G module, may be interesting from a geometric point of view. In a significant development ([20] for even characteristic and [?,?,?] for odd characteristic), Chandler, Sin and Xiang have obtained a formula for 14

17 the dimension of the q ary code generated by the characteristic functions of isotropic subspaces of a fixed dimension l in a symplectic space over F q of projective dimension n. They also identify the composition factors of the module for the corresponding symplectic group. Their method is very similar to the representation theoretic approach in [9] for the famous Hamada formula for the dimension of the q ary code generated by the characteristic functions of the subspaces of fixed dimension l in a finite projective space over F q. If n = 3 and q = p n, p any prime, F q -rank of point isotropic line incidence matrix reduces to 1 + α1 n + α2, n where α 1, α 2 = p(p+1)2 ± p(p+1)(p 1) It agrees with the expression above for p = 2. However, the methods in [20] and in [43] are different. The words of minimum nonzero weight in C are nonzero scalar multiples of 1 l, l L. (C) Subcodes of C generated by ovoids in W (q): For a subgroup A of G, let σ A denote the k endomorphism of k P taking 1 {x} k P, x P, to Σ a A 1 {a(x)} k P. If T is a subgroup of G P SP (4, q) of order q (see B.via), then σ T (l) = 1 P or 1 θi according as l S or( l is tangent to θ i for some i, 0 i q (see 2.B.ii α and v.b). Thus, σ ) T k P = 1 θ0,, 1 θq and σ T (C) = 1 θ0, 1 P C. Since G is transitive on E L and C is a kgmodule, E L C. The dimension of E L is 5 n. The composition factors of each of its socle layers is known ([43],Theorem 13, p.493). Further, E L = H L ([44],Theorem 3.1, p.5). However, the words of the minimum weight of E L are not known. If W (q) admits a polarity τ, then C contains the characteristic function 1 θ of the set θ of all absolute points of τ, a Tits ovoid of W (q). This follows because 1 θ is the diagonal of the incidence matrix of W (q) (written as (a x,y ) x,y P with a x,y = 1 or 0 according as x is, or is not, in y τ ) and the F 2 - row span of a symmetric (0, 1)- matrix contains the diagonal ([?], Theorem 3, p.143; see also [11]). Since G is transitive on T L and C is a kg- module, J L C. As noted at the end of first para of (B.vii), the characteristic function of an elliptic ovoid of W (q) is a sum of the characteristic functions of some Tits ovoids of W (q). So, E L J L C. For a proof of 1 P / J L and more on J L / E L, see [43], Theorem 15,p.495. Neither the dimension nor the words of minimum weight of J L is known. (D) On C : (i) The subcode D of C : Let D denote the set of all dual grids in W (q) (see 2. B.iv). Each word of C of minimum weight is of the form 1 m m, where m m D and m is a line of P G(3, q) not in L ([42]). 15

18 An element of D meets each element of L D in either 0, 2 or 2(q + 1) points. So, D C D. Let T and θ 0,, θ q be as in ( 2.B.v.a) and m m D. Then, by a result with R.P.Shukla (unpublished), if θ m, θ m {θ 1,, θ q } are the unique ovoids the lines m and m ( are tangent ) to, then θ m θ m. Consequently, for σ T defined in (C.iii), σ T m m = 1 θm + 1 θm which is not in σ T (C) if q > 2. If D = C, then C = D D, a contradiction if q > 2. Thus, D C if q > 2. This is as in the case of the code orthogonal to the q ary code generated by the lines of a projective 3 space over F q. However we do not know the dimension of D. Since D has only words of even weight, D E L. Does D contains 1 P \θ : θ E L? For each θ W L (q), the set of lines of P G(3, q) not in L is partitioned into the set S(θ) of secants of θ and the set E(θ) of external lines of θ. The map l l is a bijection between S(θ) and E(θ) (which does not extend linearly!) and D = {l l : l S(θ)}. Structure of the kh - submodules S(θ) and E(θ) of k P, where H is the stabilizer of θ in Aut(W L (q)), for both elliptic and Tits ovoids of W L (q). See ([47]) for a study of similar codes from ovals of Desarguesian projective planes of odd order. By (γ) in ( 2.B.ii.a), C is a one-step-completely orthogonalizable code. This means that, for each coordinate position x, its dual (that is, C) has 2 (q + 1) 1 = q + (q + 1) vectors whose supports intersect pairwise at {x} [4]. (ii) The subcode M of C : For a subgroup T of G of order q 2 + 1, let M T denote the subspace of k P of dimension q spanned by the characteristic functions of unions of even number of the T orbits θ 1,, θ q in P (see 2. B.v.a). Then, M = Σ{M T : T a subgroup of G of order q 2 + 1} k P is a kg submodule of C (see 2.B.v.b). We do not know the dimension and words of minimum weight of M. Is D M? (iii) The support of words of maximum weight in C is the compliment of an ovoid of W (q) ([3], [39]). Thus, determination of the weight enumerator of C (more to the point, the number of words of C of maximum weight, only in the case q is an odd power of two, in view of the fact that ovoids in W (4 t ) are all elliptic ([38])) settles the question of the existence of ovoids in W (q) other than the classical ovoids. We mention that there are no code words of C whose weight is in the interval (q 3 + (5q 4) /6, q 3 + q) ([39], Theorem 16, p.3137). Study of the codes O L and P \θ : θ O L may be 16

19 instructive. From the structure of C in ([43],Theorem 1, p.485), C C is the radical of C and its radical series is known. If k = F 2 and q > 2, then the weight of each element of C C is a multiple of 4, because if w C C has weight congruent to 2 (mod4), then so does σ T (w), but σ T (C) has no such element. Considering the image under σ T of sums of even number of members of {θ 1,, θ q }, we conclude that dim ( C /C C ) q 2. I do not know the simple factors of C /C C. 4 A bilinear form and a polarity on C The structure of the code C in.3 as a k[h] module, where H is the stabilizer in SP (4, q) of a quadratic form (either hyperbolic or elliptic) polarizing to a symplectic form on P G(3, q) defining SP (4, q), is determined in [45],Theorem 1. We hope that the bilinear form and a polarity on C introduced in the next section (which exist when the incidence system admits a polarity) will be helpful in understanding C, particularly its structure as a module for the stabilizer of a Tits ovoid in W (q), as well as the code over a field k of characteristic 3 associated with the (q, q)- GH H(3 2n+1 ) as a module for the Ree group 2 G 2 (3 2n+1 ). Let X = (P, L) be a finite connected partial linear space with s + 1 points on each line and s + 1 lines through each point and G = Aut(X). Assume that X admits a polarity τ, H = {g G : g τ = τ g on P L}, O = {x P : x x τ } and S = {l L : l τ l}. The sets O and S are H- invariant. The important examples here are W (q) and H(q) with q = p 2m+1 (= s), p = 2 if X = W (q) and p = 3 if X = H(q). (A) A nondegenerate bilinear form on C: Let k be a field of characteristic p; k P, k L and C be as in 1.ii and η : k L C be the surjective kg-morphism taking l L to 1 l = l C. Let B be the symmeteic k bilinear form on k L defined, for l and m L, B (l, m) to be one if l τ m and zero otherwise. Proposition 7 (i) B is H- invariant, (ii) Rad B = ker η and (iii) B induces a symmetric nondegenerate H- invariant bilinear form B on C. Proof. (i) follows from the equivalence of the following statements for l, m L and h H: B (hl, hm) = 1; (hl) τ hm; h(l τ ) hm; l τ m; B (l, m) = 1. 17

20 (ii) follows from the equivalence of the following statements for α = Σ l L λ l l, λ l k: α Rad(B ); B (α, m) = 0 for each m L; Σ m τ l Lλ l = 0 for each m L; Σ p l L λ l = 0 for each p P ; α ker η. (iii) If v, v, w, w k L are such that η (v) = η(v ) and η (w) = η(w ), then x = v v and y = w w are in ker η and B (v, w) = B (v + x, w + y) = B (v, w ), by (ii). So, B induces a symmetric H- invariant bilinear form B on C. Further, if v = Σ l L λ l l k L and w = η (v) C, then, for each m L, B(1 m, w) = B (m, Σλ l l) = Σλ l B (m, l) = Σ m τ lλ l = w mτ (**) where, for p P, w p denotes the p th -coordinate of w. As τ is a bijection between P and L, this implies that B(1 m, w) = 0 for each m L if, and only if, w = 0. Thus, B is nondengererate. Note that the radical of the restriction of f (defined in 1.i) to C is C C, which unlike radical of B, may be nonzero. For any subspace D of C, we denote by D the orthogonal of D with respect to B. If (p, s + 1) = 1, then 1 P = Σ l L 1 l C and, for w C, by (**), B(w, 1 P ) = B(w, Σ l L 1 l ) = Σ l L w l τ = Σ p P w p = f(w, 1 P ). So, 1 P = 1 P = C0. For x P, let π x = Σ x m L m k L and π x = η ( π x ) C. Then, for w C, B(π x, w) = Σ p x τ w p (by (**)) = f (w, x τ ). So, for Π = π x : x P C, Π = C C. The subspaces M = {v C : B(v, 1 m ) = f(v, 1 m ) for each m L}, U = 1 l : l S and W = 1 l : l L \ S are kh- submodules of C. An element w C is in M if, and only if, w m τ = Σ p m τ w p for each m L. So, 1 P M. Let p = 2 and write w C as w = Σ l S λ l 1 l + Σ l L\S µ l 1 l, λ l, µ l k. Then, B(w, w) = Σ l S λ 2 l = (Σ l Sλ l ) 2. So, the set of all isotropic elements of C with respect to B is U 0 + W. Use of ([3], Theorem 3, p.143) again yields 1 O C. Since x O if, and only if, x τ S, by (**), B(1 m, 1 O ) = 1 if m τ O and zero otherwise. So, 1 O (U 0 + W ). Since U 0 + W is of codimension zero in C if O is empty and one otherwise, (U 0 + W ) = k1 O. (B) A polarity of C: Let θ : k P C be the surjective kh module homomorphism taking x P to τ(x). Recall A defined for A k P. Proposition 8 (i) 0 C k P θ C 0 is an exact sequence of kh modules. (ii) For any subspace A of C, dim A+ dim θ(a ) = dim C. Proof. (i) By definition, θ is onto. (i) follows from the equivalence of the following statements for v = Σ x P v x x k P : v ker θ; Σ x P v x 1 x τ = 0; 18

21 Σ{v x : x P such that p x τ } = 0 for each p P ; Σ x p τ v x = 0 for each p P ; Σ x l v x = 0 for each l L; Σ x P v x x C. (ii) dim A+ dim A = P = dim C+ dim C. Therefore, dim A+ dim A dim C = dim C. But C A and A /C θ(a ). So, (ii) holds. Let P (V ) denote the set of all subspaces of a vector space V, partially ordered by inclusion. Consider the maps P(k P ) P(k P ) θ P(C), where, for A k P, takes A to its orthogonal compliment A with respect to f and, if A P(k P ), θ (A) = θ (A). We write A = θ ( A ). Clearly, if A is a kh submodule of k P, then so is A. Proposition 9 Let A P(k P ). (a) C A if, and only if, A = 0. (b) If A P(C), then (i) dim A+ dim A = dim C; and (ii) A = A. (c) The map A A, A P(C), is an inclusion reversing involutory permutation of P(C). (d) If (p, s + 1) = 1, then 1 P C, (k1 P ) = C 0 and (A ) 0 = (A+k1 P ) for each subspace A of C. In particular, the map in (c) is an inclusion reversing bijection between the set of all subspaces of C containing 1 P and the set of all subspaces of C 0. Proof. Since a subspace A contains C if, and only if, A C, (a) follows from Proposition 8 (i). (b. i) is a restatement of Proposition 8 (ii). Since the dimensions of A and A are equal (by (b.i)), we need only to show that A A. Note that A = { Σ p P (Σ x p τ β x ) p : Σ p P β p p A }. Consider w = η(σ l L λ l l) C. Then, w = Σ x P (Σ x l L λ l ) x. Now, Σ l L λ l l τ ( A ) if, and only if, for each w = Σβ x x A, 0 = Σ l τ P λ l (Σ l τ x τ β x) = Σ l τ P λ l (Σ x l β x ) = Σ x P (Σ x l L λ l ) β x = f(w, w ). 19

22 So, Σ l L λ l l τ ( A ) if, and only if, w A = A. Since w = θ (Σ l L λ l l τ ), it follows that A = A. So, (b.ii) follows. (c) is now follows from (b). For w = Σ x P λ x x k P, θ (w) = η(σ x P λ x x τ ) and c(θ(w)) = (s + 1)c(w). Since Σ l L 1 l = (s + 1) 1 P C and (p, s + 1) = 1, 1 P C. For any subspace A of k P, θ(a 1 P ) = θ (A) 1 P C0. So, (k1 P ) = θ(k1 P ) C0. Since C 0 and (k1 P ) are both of codimension one in C (see Proposition 9 (b.i)), (k1 P ) = C 0. For A P(C), A 1 P = (A + k1 P ). So, (A ) = θ ( A ) 1 P = θ(a 1 P ) = θ((a + k1 P ) ) = (A + k1 P ). So, a subspace A of k P contains 1 P if, and only if, A C 0. So, (d) follows. Let l L. If w = Σ p P β p p k P, then w l if, and only if, Σ p l β p = 0. In this case, l τ -th coordinate of θ (w) is zero. So, θ(k1 l ) {w C : W L (q) = 0}. Since both subspaces are of codimension one in C, equality holds. As noted above, (k1 P ) = C 0. When (p, s + 1) = 1, the polarity of P (C) taking A P (C) to A is not symplectic, because 1 P / (k1 P ) = C 0. References [1] T.L. Alderson and K.E. Mellinger, Partitions in finite geometry and related constant composition codes, Innov. Incidence Geom. 8 (2008), [2] O. Arslan and P. Sin, Some simple modules for classical groups and p ranks of orthogonal and Hermitian geometries, J. of Algebra 327 (2011) [3] B. Bagchi and N.S.N. Sastry, Even order inversive planes, generalized quadrangles and codes, Geom. Dedicata 22 (1987) [4] B. Bagchi and N.S.N. Sastry, One step completely orthogonalizable codes from generalized quadrangles, Information and computation, 77 (1988) [5] B. Bagchi and N.S.N. Sastry, Intersection pattern of the classical ovoids in symplectic 3-space of even order, J. of Algebra 126 (1989)

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