De Beule, Jan

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Ghent University Academic Bibliography

An application of {δ(q + 1), δ;n+ 1, q}-minihypers on generalizedquadranglesJ. De Beule(joint work with M. R. Brown and L. Storme)AbstractMinihypers in finite projective spaces have been used greatly tostudy the problem of linear codes meeting the Griesmer bound; therebyshowing their importance for coding theory. But they are also impor-tant for a great variety of geometrical problems. Using the classifica-tion of {δ(q+1), δ;n+1, q}-minihypers we obtain results on spreads ofcertain finite generalized quadrangles. We discuss both the applicationand the result.1 IntroductionIn this section we introduce the concept of a generalized quadrangle, orshortly, a GQ.Definition 1.1 A (finite) generalized quadrangle (GQ) is an incidence struc-ture S = (P,B, I) in which P and B are disjoint non-empty sets of objectscalled points and lines (respectively), and for which I⊆ (P ×B)∪ (B×P) isa symmetric point-line incidence relation satisfying the following axioms:(i) Each point is incident with 1 + t lines (t > 1) and two distinct pointsare incident with at most one line.(ii) Each line is incident with 1 + s points (s > 1) and two distinct linesare incident with at most one point.(iii) If x is a point and L is a line not incident with x, then there is aunique pair (y,M) ∈ P × B for which x I M I y I L.The integers s and t are the parameters of the GQ and S is said to haveorder (s, t). If s = t, then S is said to have order s.Examples: Classical examples are the symplectic space W3(q) in PG(3, q),the hyperbolic quadric Q+(3, q) in PG(3, q), the parabolic quadric Q(4, q)in PG(4, q), the elliptic quadric Q−(5, q) in PG(5, q), and the Hermitianvarieties H(3, q2) and H(4, q2) in PG(3, q2) and PG(4, q2), which have re-spectively order q, (q, 1), q, (q, q2), (q2, q) and (q2, q3). The non-classicalexamples of Tits are given in the following definition:Definition 1.2 Let n = 2 (respectively, n = 3) and let O be an oval (re-spectively, an ovoid) of PG(n, q). Furthermore, let PG(n, q) be embedded asa hyperplane in PG(n+ 1, q).Define points as1(i) the points of PG(n+ 1, q) \ PG(n, q),(ii) the hyperplanes X of PG(n+ 1, q) for which |X ∩ O| = 1, and(iii) one new symbol (∞).Lines are defined as(a) the lines of PG(n+1, q) which are not contained in PG(n, q) and whichmeet O (necessarily in a unique point), and(b) the points of O.Incidence is inherited from PG(n+ 1, q), whereas the point (∞) is incidentwith no line of type (a) and with all lines of type (b).It is straightforward to show that these incidence structures are GQ’s withparameters s = q, t = qn−1.Definition 1.3 A spread of a GQ S of order (s, t) is a set S of lines suchthat every point of S is incident with exactly one element of S. A spreadnecessarily contains 1 + st lines. A partial spread is a set S of lines forwhich every point is incident with at most one line of S. A partial spread iscalled maximal if S is not contained in a larger partial spread. If the size ofa partial spread is 1 + st− δ, then δ is said to be the deficiency of the partialspread.The natural question is whether a partial spread with certain deficiency canbe maximal, or, in other words, can a partial spread with small deficiencybe extended? Using minihypers we can give answers to this question forpartial spreads of the GQ’s T2(O) and T3(O).2 The minihypersDefinition 2.1 An {f,m;N, q}-minihyper is a pair (F,w), where F is asubset of the point set of PG(N, q) and where w is a weight function w:PG(N, q)→ N: x 7→ w(x), satisfying1. w(x) > 0 ⇐⇒ x ∈ F ,2.∑x∈F w(x) = f , and3. min{∑x∈H w(x)‖H ∈ H} = m, where H is the set of hyperplanes ofPG(N, q).Related to certain minihypers are blocking sets of PG(2, q). The followingtheorem about blocking sets is used for the final theorem.2Theorem 2.2 (A. Blokhuis, L. Storme, and T. Szo˝nyi [3]) Let B be a block-ing set in PG(2, q), q = ph, p prime, of size q + 1 + c. Let c2 = c3 = 2−1/3and cp = 1 for p > 3.1. If q = p2d+1 and c < cpq2/3, then B contains a line.2. If 4 < q and q is a square and c < cpq2/3, then B contains a line or aBaer subplane.To establish the connection between blocking sets and certain minihyperswe need one more definition.Definition 2.3 Let A be the set of all lines of PG(N, q). A sum of linesis a weight function w: A → N: L 7→ w(L). A sum of lines inducesa weight function on the points of PG(N, q), which is given by w(x) =∑L∈A,x∈Lw(L). In other words, the weight of a point is the sum of theweights of the lines passing through that point. A sum of lines is said to bea sum of n lines if the sum of all the weights of the lines is n.The connection is finally expressed in the following theorem, which will beof direct use for our applicationTheorem 2.4 (Govaerts and Storme [1]) Let (F,w) be a {δ(q+1), δ;N, q}-minihyper, q > 2, satisfying 0 6 δ < , where q+  is the size of the smallestnon-trivial blocking set in PG(2, q). Then w is a weight function induced onthe points of PG(N, q) by a sum of δ lines.3 The applicationConsidering an arbitrary partial spread of Tn(O), we will define an {δ(q +1), δ;n+ 1, q}-minihyper.Definition 3.1 Let S be a partial spread of a GQ. A hole with respect to Sis a point of the GQ which is not incident with any line of S.Consider a partial spread S of Tn(O), n = 2 or n = 3, of size qn + 1 −δ. Referring to the definition of the GQ Tn(O), let pi0 = PG(n, q) whichcontains O and which is embedded in PG(n + 1, q) as a hyperplane. Weremark that a partial spread contains at most one line of type (b) of theGQ, because all lines of type (b) intersect in (∞).Definition 3.2 Let S be a partial spread of Tn(O) (n = 2 or n = 3). DefinewS: PG(n+ 1, q)→ N as follows:(i) if x ∈ PG(n+1, q)\pi0 and x is a hole with respect to S, then wS(x) = 1,otherwise wS(x) = 0,3(ii) suppose x ∈ O, define wS(x) = δx, with q − δx the number of lines ofS through x.(iii) wS(x) = 0, ∀x ∈ pi0 \ O.This weight function determines a set F of points of PG(n+ 1, q). We willdenote the defined minihyper by (F,wS).We can now proveLemma 3.3 Let S be a partial spread of Tn(O) (n = 2 or 3) which covers(∞) and which has deficiency δ < q. Then wS is the weight function of a{δ(q + 1), δ;n+ 1, q}-minihyper (F,wS).This lemma leads immediately toTheorem 3.4 (M.R. Brown, J. De Beule and L. Storme [2]) Let S be apartial spread with deficiency δ of Tn(O) (n = 2 or 3) covering (∞). Ifδ < , with q+  the size of the smallest non-trivial blocking set in PG(2, q),q > 2, we can always extend S to a spread.References[1] P. Govaerts and L. Storme. On a particular class of minihypers and itsapplications. I. The result for general q. Des. Codes Cryptogr., accepted.[2] M.R. Brown, J. De Beule and L. Storme. Partial spreads of T2(O) andT3(O). European J. Combin., accepted.[3] A. Blokhuis, L. Storme and T. Szo˝nyi. Lacunary polynomials, multipleblocking sets and Baer subplanes. J. London Math. Soc. (2), 60(2):321–332, 1999.Address of the authors: Ghent University, Dept. of Pure Maths andComputer Algebra, Krijgslaan 281, 9000 Gent, Belgium(M.R. Brown: mbrown@cage.rug.ac.be)(J. De Beule: jdebeule@cage.rug.ac.be, http://cage.rug.ac.be/∼jdebeule)(L. Storme: ls@cage.rug.ac.be, http://cage.rug.ac.be/∼ls)4

An application of {delta(q+1),delta;n,q}-minihypers on generalized quadrangles

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An application of {delta(q+1),delta;n,q}-minihypers on generalized quadrangles

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