Designs in finite classical polar spaces

The classification of non-degenerate sesquilinear and non-singular quadratic forms on vector spaces over a finite field gives rise to the finite classical polar spaces. Fixing such a form on a finite ambient vector space V, the polar space \({\mathcal {Q}}\) is formed by all subspaces of V which are totally isotropic or totally singular. The subspaces of maximal dimension are called generators and their dimension is called the rank of \({\mathcal {Q}}\). The type of the form together with the order q of the base field will be called the type Q of \({\mathcal {Q}}\).

A set \({\mathcal {S}}\) of generators of \({\mathcal {Q}}\) such that each point of \({\mathcal {Q}}\) is incident with exactly \(\lambda \) elements of \({\mathcal {S}}\) is called a \(\lambda \)-spread. The study of these objects goes back to Segre (1965) [30]. Segre’s work already contains a generalization replacing the points by subspaces of arbitrary dimension t, known as a \(\lambda \)-regular system with regard to \((t-1)\)-spaces. This definition clearly is related to the notion of combinatorial designs [12] and subspace designs [7], where the ambient subset (or subspace) lattice has been replaced by the subspaces of the polar space \({\mathcal {Q}}\), and it is covered by Delsarte’s designs in association schemes, see Vanhove [32, pp. 39, 81, 193].

This naturally leads us to the following full adaption of the notion of a design to polar spaces. A t-\((r, k, \lambda )_Q\) design is a set D of k-dimensional subspaces of the ambient polar space \({\mathcal {Q}}\) of type Q such that every t-dimensional subspace of \({\mathcal {Q}}\) is incident with exactly \(\lambda \) elements (called blocks) of D. A design with \(\lambda = 1\) is called Steiner system. The \(\lambda \)-regular systems with regard to the \((t-1)\)-spaces form the special case \(k = r\).

Apart from the elementary cases of trivial designs and designs coming from Latin–Greek halvings, hitherto only very few concrete constructions of designs in polar spaces of strength \(t\ge 2\) have been known. De Bruyn and Vanhove (2012) constructed designs with parameters 2-\((3,3,2)_{\Omega (3)}\) and 2-\((3,3,2)_{\Omega ^-(2)}\), and showed the non-existence of 2-\((3,3,2)_{{{\,\textrm{Sp}\,}}(3)}\) design [15]. In the sequel, Bamberg and Lansdown [3, 26] constructed 2-\((3,3,(q+1)/2)_{\Omega (q)}\) designs for \(q\in \{5,7,11\}\). Non-existence results for Steiner systems in polar spaces have been shown in [29], see Theorem 20. By the recent non-constructive result [35], nontrivial t-\((r,k,\lambda )_Q\) designs exist for all values of t and all types Q, provided that \(k > \frac{21}{2} t\) and r is large enough.

In this article, we investigate the theory of classical and subspace designs for their applicability to designs in polar spaces. After collecting the required preliminaries in Sect. 2, Sect. 3 states the definition of a design D in a polar space and gives the first basic results, including the supplementary (Lemma 4) and the reduced (Lemma 5) design of D. The latter leads to the notion of admissible parameters. In Sect. 4, we construct derived (Theorem 6) and residual (Theorem 7) designs of D. Equations for the intersection numbers of D are computed in Sect. 5 (Theorems 11 and 12) and yield a uniqueness result for the Latin–Greek halvings (Lemma 14) in hyperbolic polar spaces. In Sect. 6, we show that the Gram matrix of the point-block incidence matrix of a design D of strength \(t \ge 2\) is related to the adjacency matrix of the collinearity graph of the ambient polar space. This leads to an analog of Fisher’s inequality (Theorem 18), which in turn yields the classification of all t-\((r,r,1)_{\Omega ^+(q)}\) Steiner systems of strength \(t \ge 2\) (Theorem 19) by elementary means. In the computational part in Sect. 7, we construct designs with \(t=2\) for more than 140 previously unknown parameter sets in various classical polar spaces over \(\mathbb {F}_2\) and \(\mathbb {F}_3\). Among these are the first nontrivial 2-designs where the block dimension k is strictly smaller than the rank r of the ambient polar space. We conclude the article stating a few open problems and suggestions for further research in Sect. 8.

Throughout this article, \(q \ge 2\) will denote a prime power and V a vector space over \(\mathbb {F}_q\) of finite dimension v. The lattice \({{\,\textrm{PG}\,}}(V) \cong {{\,\textrm{PG}\,}}(n-1,q)\) of all subspaces of V is a finite projective space of algebraic dimension n and geometric dimension \(n-1\). For subspaces of projective and polar spaces, the word dimension will always denote the algebraic dimension, as the resulting formulas tend to be simpler and closer to the classical theory of block designs. The set of all \(\mathbb {F}_q\)-subspaces of V of dimension k will be denoted by . Its cardinality is given by the Gaussian binomial coefficientWe will make use of the abbreviation , which is known as the q-analog of the number v. The subspaces of algebraic dimension 1 will be called points, of dimension 2 lines, of dimension 3 planes and dimension \(v - 1\) hyperplanes.

For a subspace U of V of dimension u, V/U is the ambient vector space of a projective space over \(\mathbb {F}_q\) of dimension \(v - u\). As a consequence, the number of subspaces of V of dimension k containing U is .

For an introduction to finite polar spaces, we refer to [2, 14]. Table 1 shows all classical polar spaces of rank \(r \ge 1\) over the finite field \(\mathbb {F}_q\) up to isomorphism. It will be convenient to collect all finite classical polar spaces \(\mathcal{Q}\) which only differ (possibly) in their rank r (but not in the type of the underlying form nor the number q) into the type Q. The table includes the symbol we will use for the type, as well as a parameter \(\epsilon \) which allows a uniform treatment of all types of polar spaces in counting formulas. Note that the Hermitian polar spaces only exist for squares q. For even q, there is an isomorphism \(\Omega (2t+1) \cong {{\,\textrm{Sp}\,}}(2r,q)\). For the readers’ convenience, the second last column lists an alternative set of symbols that are sometimes found in the literature (for example [29]). Each polar space has a natural embedding into a projective geometry \({{\,\textrm{PG}\,}}(V)\). The column n lists the (algebraic) dimension \(\dim (V)\) of this geometry, and column G contains the collineation group, see e.g. [9, Section 2.3.5].

In the following, \(\mathcal{Q}\) will always denote a finite classical polar space over \(\mathbb {F}_q\) of rank \(r \ge 1\), Q will denote its type and \(\epsilon \) will be its parameter as listed in Table 1. Corresponding to the notation for vector spaces, the set of all subspaces of \(\mathcal{Q}\) of dimension k will be denoted by , and its cardinality will be denoted by . Two distinct points are called collinear if the line \(\langle P,P'\rangle \) is contained in \({\mathcal {Q}}\). For a subspace U of the polar space \(\mathcal{Q}\), the symbol \(U^\perp \) denotes the orthogonal subspace of U in V with respect to the underlying form. Note that \(\dim (U) + \dim (U^\perp ) = n\), and for all subspaces U of \({\mathcal {Q}}\) we have \(U \subseteq U^\perp \).

A hyperplane H of the ambient vector space V is either degenerate or non-degenerate with respect to the underlying form. The degenerate hyperplanes are exactly the ones of the form \(H = P^\perp \) with . Moreover, all subspaces of \(\mathcal{Q}\) contained in H pass through the point P, and after modding out P, the resulting set of spaces is a polar space of the same type Q and rank one less. For non-degenerate hyperplanes H, the restriction of \(\mathcal{Q}\) to H is again a polar space.

We need some basic counting formulas in the subspace poset of finite classical polar spaces. A useful observation is that for a given subspace U of dimension u in the poset of all subspaces of the finite polar space \(\mathcal{Q}\), the interval below U is the finite projective space \({{\,\textrm{PG}\,}}(U) \cong {{\,\textrm{PG}\,}}(u-1,q)\), and the interval above U is the finite polar space \(U^\perp / U\), which is of the same type Q and of rank \(r - u\). Therefore, counting subspaces below U just resembles counting in finite projective spaces as discussed above. For counting subspaces above U, we prepare the following lemma.

Again, we define the abbreviation . Table 2 lists the polar spaces \(\mathcal{Q}'\) arising from non-degenerate hyperplanes of the ambient vector space of \(\mathcal{Q}\), together with the rank \(r'\) and the parameter \(\epsilon '\) of \(\mathcal{Q}'\). Note that in all cases, \(r - r' \in \{0,1\}\) and \(\epsilon ' = \epsilon + 2(r-r') - 1\). We remark that the symplectic space \(\mathcal{Q}= {{\,\textrm{Sp}\,}}(2r,q)\) does not show up in the table, since its point set equals the full point set of \({{\,\textrm{PG}\,}}(V)\) (as all vectors are isotropic) and therefore, all hyperplanes are degenerate.

The collinearity graph of a polar space \(\mathcal{Q}\) has the points of \(\mathcal{Q}\) as vertex set, with two of them being adjacent if they are collinear. This graph is known to be strongly regular with the parameters \((v,k,\lambda ,\mu )\) where [18]From these parameters, the eigenvalues can be computed as \(\theta _0 = k\) with multiplicity \(m_0=1\) and the two zeroes \(\theta _1\), \(\theta _2\) of the quadratic equation \(\theta ^2 - (\lambda - \mu )\theta - (k - \mu ) = 0\). The remaining multiplicities \(m_1\) and \(m_2\) are then determined by the system of linear equations \(\theta _1 m_1 + \theta _2 m_2 = -1\) and \(m_1 + m_2 = v - 1\). This leads [9, Theorem 2.2.12] to the eigenvalueswith multiplicities

Since the early 1970s, the notion of combinatorial block designs has been generalized to subspace designs, which are designs in finite projective spaces and can be understood as a q-analog of the classical situation. Many results in classical block designs turned out to have a generalization for subspace designs. We refer the reader to [12] for a comprehensive treatment of combinatorial designs and [7] for an introduction and an overview to subspace designs.

The theory of subspace designs will serve as a blueprint for our investigation of designs in finite classical polar spaces. As we will see, parts of the basic theory can also be applied in this situation, while others apparently do not have a direct adaption. Again, \(\mathcal{Q}\) will denote a finite classical polar space over \(\mathbb {F}_q\) of rank \(r \ge 1\), Q will denote its type and \(\epsilon \) will is its parameter as listed in Table 1.

We remark that our definition doesn’t allow repeated blocks, so in design theory terminology, all designs are considered to be simple. Steiner systems with \(t = 1\) are called spreads and are well-studied objects, compare [14].

In finite geometry, designs with parameters t-\((r,r,\lambda )_Q\), i.e. the dimensions of the blocks equal the rank of the polar space, were already studied by Segre [30] under the name \(\lambda \)-regular systems with respect to \((t-1)\)-spaces. \(\lambda \)-regular systems with respect to points are called \(\lambda \)-spreads, see [13] and [9, Sec. 2.2.7].

Of course, the empty design is a t-\((r,k,0)_q\) design for all values \(t\in \{0,\ldots ,k\}\).

The following two lemmas are a direct consequence of Lemma 1(b).

By the last lemma, only the range \(\lambda \in \{1,\ldots , \lfloor \lambda _{\max } / 2\rfloor \}\) is relevant for the investigation of designs. Designs D with \(\lambda = \lambda _{\max }/2\) are called halvings as they split into two “halves” D and \(D^\complement \) which both are designs with the same parameters. In the case \(r = k\) halvings are also called hemisystems [13].

For \(s = t-1\), the resulting design of Lemma 5 is called the reduced design of D. The number \(\lambda _1\) is called the replication number of D. We remark that \(\lambda _t = \lambda \). For a t-\((r,k,\lambda )_{Q}\) design, the numbers \(\lambda _0, \ldots ,\lambda _t\) clearly must be integers. If these integrality conditions are satisfied, the parameters t-\((r,k,\lambda )_{Q}\) will be called admissible. If for admissible parameters a design exists, the parameters are called realizable.

For fixed Q, r, k and t, the set of the admissible parameters will have the form t-\((r,k,\lambda )_Q\) with \(\lambda \in \{0,\Delta _\lambda ,2\Delta _\lambda ,\ldots ,\lambda _{\max }\}\), whereIn the important case \(k = r\) we always get \(\Delta _\lambda = 1\), implying that the integrality conditions do not give any further restrictions on the parameters.

Now we investigate the restriction of designs to hyperplanes H. Because of [20], resulting constructions can be seen as a polar space analog of the derived (Theorem 6) and the residual (Theorem 7) design.

We start with the easier case of a degenerate hyperplane H. Here \(H = P^\perp \) with and we want to mod out P to get a polar space.

In the theorem, the possible combinations for \(({\mathcal {Q}},\bar{{\mathcal {Q}}})\) are \((\Omega ^-(2r+2,q),\Omega (2(r+1) + 1,q))\), \((\Omega (2r + 1,q),\Omega ^+(2(r+1),q)\) and \((U(2r+1,q),U(2(r+1),q))\). The construction is based on the fact that in these situations, every generator of \(\bar{{\mathcal {Q}}}\) contains a unique generator of \({\mathcal {Q}}\). We remark that our residual design in Theorem 7 is the derived design of \(\bar{D}\) in Theorem 9.

So far, we have analogs of the supplementary, the reduced, the derived and the residual design in polar spaces. In the theory of block and subspace designs, there is a further general construction, the dual (or complementary) design. However, we did not find a convincing counterpart of that construction. Our feeling is that there might not be such a thing, since the classical construction of the dual design is based on the fact that the underlying subset resp. subspace lattice is self-dual, which however is not true for the poset of all subspaces of a polar space.

Another classical topic in the theory of combinatorial and subspace designs are the intersection numbers, which describe the intersection sizes of the blocks of a design with a fixed subspace S. For combinatorial designs they have been originally defined in [27] for blocks S and independently as “i-Treffer” for general sets S in [28].

Lemma 5 allows us to develop the corresponding result for designs in polar spaces. In fact, the statements and the proofs can be done almost literally in the same way as for subspace designs in [21].

For a fixed t-\((r,k,\lambda )_Q\) design D in the polar space \({\mathcal {Q}}\) and a subspace S of \({\mathcal {Q}}\), we define the i-th intersection number (\(i\in \{0,\ldots ,k\}\)) asIf the space S is clear from the context, we use the abbreviation \(\alpha _i = \alpha _i(S)\).

First, we derive a polar space counterpart of the intersection equations or Mendelsohn equations, see [27, Th. 1], [28, Satz 2] for classical designs and [21, Th. 2.4] for subspace designs.

The intersection equations can be read as a linear system of equations \(Ax = b\) on the intersection vector \(x = (\alpha _0, \alpha _1,\ldots , \alpha _k)\). The left \((t+1)\times (t+1)\) square part of the matrix \((i\in \{0,\ldots ,t\}\) and \(j\in \{0,\ldots ,k\})\) is called the upper triangular q-Pascal matrix, which is known to be invertible with inverse matrix . Left-multiplication of the equation system with this inverse yields a parameterization of the intersection numbers \(\alpha _0,\ldots ,\alpha _t\) by \(\alpha _{t+1},\ldots ,\alpha _k\). In this way, we get a counterpart of the Köhler equations, see [24, Satz 1] for classical designs and [21, Th. 2.6] for subspace designs.

In hyperbolic polar spaces \(\Omega ^+(q)\), there is a unique partition of the generators into two parts, commonly called Latins and Greeks, which are a design with the parameters \((r-1)\)-\((r,r,1)_{\Omega ^+(q)}\), see e.g. [14]. We will call them legs of the Latin–Greek halvings, or simply the Latin–Greek halvings. (Note that by \(\lambda _{\max } = q^{1+\epsilon } + 1 = 2\), they are halvings.) For two generators B and \(B'\), the number \(r - \dim (B \cap B')\) is even if and only if B and \(B'\) are contained in the same leg of the Latin–Greek halving. Hence the intersection numbers of the Latin–Greek halving with respect to a block S have the property \(\alpha _i = 0\) if \(r-i\) is odd.

The theory of intersection numbers yields the following uniqueness result.

For combinatorial designs, a classical result is Fisher’s inequality [4, 12], which states that a t-\((v,k,\lambda )\) design of strength \(t \ge 2\) has at least as many blocks as points. In the case of equality, a design is called symmetric. For subspace designs, Fisher’s inequality has been proven in [11], also showing that there are no symmetric subspace designs of strength \(t \ge 2\). For generalizations of Fisher’s inequality for classical and subspace designs, see [22].

The natural question in our situation is if there exist designs such thatMoreover, we would also like to investigate the equality case, where the corresponding designs are called symmetric. We start by looking at the consequences of Lemma 5.

The common textbook proof of Fisher’s inequality for classical and subspace designs uses Bose’s elegant argument based on the rank of the point-block incidence matrix N [4]. We follow this approach for our case of designs in polar spaces.

For a combinatorial 2-\((v,k,\lambda )\) design (or a subspace design) it is easy to see thatwhere J is the all-one matrix. Now, let D be a 2-\((r,k,\lambda )_Q\) design and let N be the point-block incidence matrix of D. Then for two points we haveIt follows thatwhere A is the adjacency matrix of the collinearity graph of the polar space.

The equation for \(NN^\top =\lambda _1 I + \lambda \cdot A\) for designs in polar spaces is a straightforward generalization of the equation \(NN^\top = (\lambda _1 -\lambda )I + \lambda J\) for subspace designs, since \(J-I\) can be regarded as the adjacency matrix of the collinearity graph, i.e. the complete graph, in PG(V, q).

For \(k < r\), Fisher’s inequality follows already from Lemma 15. However, for \(k = r\), Fisher’s inequality (18) sharpens Lemma 15, which leads to the following classification result.

An important part of our proof of Theorem 19 is to show the non-existence of t-\((r,r,1)_{\Omega ^+(q)}\) designs with \(t \ge 2\) and \(r \ge 4\). This result follows also from non-existence stated in Theorem 20 below, which has been proven by Schmidt and Weiß using the machinery of association schemes. Our argument based on Fisher’s inequality provides an elementary proof of that special case.

For other non-existence results see [13]. As t-\((r,r,\lambda )_Q\) designs are t-designs in association schemes (compare [32, 13, Prop 2.2]), Delsarte’s general theory can be applied and gives the following stronger version of Fisher’s inequality.

For polar spaces, the multiplicities \(\mu _i\) have been determined by Stanton [31, Cor. 5.5]. For \(t=2,3\), Delsarte’s theorem gives \(\# D \ge 1 + \mu _1\) and \(\mu _1\) equals the multiplicity \(m_1\) of one of the eigenvalues of the collinearity graph in Theorem 18.

K.-U. Schmidt was confident that Delsarte’s inequality for designs in polar spaces could be extended to \(k < r\), too. Moreover, Delsarte’s inequality is a special case of the linear programming bound for designs in association schemes, and he conjectured that the linear programming bound allows to extend the non-existence results in Theorem 20 from \(\lambda =1\) to something like \(\lambda \le q^{\left( {\begin{array}{c}t\\ 2\end{array}}\right) }\). This conjecture is supported by our computational results in Sect. 7.1. Unfortunately, after his tragic, untimely death his latest results in this regard seem to be lost.

For the construction of designs we use the well-known approach to choose a group G and search for G-invariant designs. Thus, instead of searching for individual k-subspaces the design has to be composed of G-orbits on the k-subspaces. This approach applied to combinatorial designs is commonly attributed to Kramer and Mesner [25] and has been used there with great success.

For a subgroup G of the collineation group of a polar space \(\mathcal{Q}\), the set of blocks of a G-invariant t-\((r,k,\lambda )_Q\) design (\(G\le {{\,\textrm{Aut}\,}}(\mathcal{Q})\)) is the disjoint union of orbits of G on the set of k-dimensional subspaces of \(\mathcal{Q}\). To obtain an appropriate selection of orbits of G on we consider the incidence matrix \(A_{t,k}^G\) whose rows are indexed by the G-orbits on the set of t-subspaces of \(\mathcal{Q}\) and whose columns are indexed by the orbits on k-subspaces. The entry \(a_{T,K}^G\) of \(A_{t,k}^G\) corresponding to the orbits \(T^G\) and \(K^G\) is defined byNow the theorem by Kramer and Mesner, originally formulated for combinatorial designs, can be applied to designs in polar spaces as well:

The question now is which are the promising groups that can be prescribed in Theorem 22? For combinatorial designs, cyclic groups have turned out to be promising candidates for prescribed groups in Theorem 22. For subspace designs, the Singer cycle and its normalizer, and subgroups thereof are good candidates [8]. For designs in polar spaces, it is not yet clear if there are exceptionally promising groups to be used for Theorem 22.

With these previous experiences in mind, we constructed all cyclic subgroups of the automorphism group of each polar space together with its normalizer using MAGMA [5]. The cyclic groups and random subgroups of their normalizers were prescribed as automorphism groups in Theorem 22.

The systems of Diophantine linear equations (2) were attempted to be solved by the third author’s software solvediophant [33, 34]. The parameters for which designs have been found are listed in Sect. 7.1. In small cases, exhaustive non-existence results could be achieved. So far, no designs could be found in the Hermitian polar spaces. However, this seems rather to be due to the huge size of the search space than due to the scarcity of designs.

We would like to conclude this article by pointing out some open questions and suggestions for future work.

In the theory of combinatorial and subspace designs, the numbers \(\lambda _{i,j}\) of a t-design D play an important role. They can be defined in two variants (which coincide in the case of combinatorial designs). Let i, j be non-negative integers with \(i+j \le t\). For subspaces I, J with \(\dim (I) = i\) and \(\dim (J) = j\), the number of blocks \(B \in D\) containing I and having trivial intersection with J only depends on i and j and yields the first variant of \(\lambda _{i,j}\). Similarly, for subspaces I, J with \(\dim (I) = i\) and \({{\,\textrm{codim}\,}}(J) = j\), the number of blocks \(B \in D\) with \(I \subseteq B \subseteq J\) only depends on i and j, which gives the second variant. It might be interesting to investigate this concept for designs in polar spaces.

For combinatorial and subspace designs, a large set is a partition of the set of all k-subspaces of the ambient space into N t-designs of the same size. Large sets generalize the notion of a halving (and thus of hemisystems in polar spaces), which are given by the case \(N = 2\). An important application of large sets of combinatorial and subspace designs is the construction of infinite series of designs, see [6, 19]. Maybe this strategy can also be applied to large sets of designs in polar spaces.

One could further investigate symmetric 2-\((4,4,q+1)_{\Omega ^+(q)}\) designs, which are the only parameters where the symmetric designs in polar spaces have not been fully classified in Sect. 6. Moreover, we would like to ask if inequality (1) allows any designs in \(U^+(q)\), and if it allows any non-complete and non-Latin–Greek designs in \(\Omega ^+(q)\).

For q odd, it is an open question whether a spread in \(\Omega ^-(2r+2, q)\), \(r\ge 3\) exists [17, Sec. 7.5]. In our notation, these are the designs with the parameters 1-\((r, k, 1)_{\Omega ^-(q)}\). By computer search, we could show that if in \(\Omega ^-(8, 3)\) such a spread exists, the only prime powers that may divide the order of an automorphism are 2 or 3.

An m-dimensional Kerdock set over \(\mathbb {F}_q\) with even m is a set of \(q^{m-1}\) skew-symmetric \((m\times m)\)-matrices such that the difference of any two of them is non-singular. Its existence is equivalent to the existence of a 1-\((m,m,1)_{\Omega ^+(q)}\) spread. It has been shown that for even q, the Kerdock sets do exist for all m. For odd q, the existence for \(m \ge 6\) is open. We could show by a computer search that if a 1-\((6,6,1)_{\Omega ^+(3)}\) exists, its automorphism group is of order 1, 2, 3, 4, 5, 6 or 8.