q-analogs of group divisible designs

-analogs of group divisible designs Marco Buratti, Michael Kiermaier, Sascha Kurz, Anamari Nakić, Alfred Wassermann September 7, 208 Abstract A well known class of objects in combinatorial design theory are group divisible designs. Here, we introduce the -analogs of group divisible designs. It turns out that there are interesting connections to scattered subspaces, -Steiner systems, design packings and r -divisible projective sets. We give necessary conditions for the existence of -analogs of group divsible designs, construct an infinite series of examples, and provide further existence results with the help of a computer search. One example is a (6, 3, 2, 2) 2 group divisible design over GF(2) which is a design packing consisting of 80 blocks that such every 2-dimensional subspace in GF(2) 6 is covered at most twice. Introduction The classical theory of -analogs of mathematical objects and functions has its beginnings as early as in the work of Euler [Eul53]. In 957, Tits [Tit57] further suggested that combinatorics of sets could be regarded as the limiting case of combinatorics of vector spaces over the finite field GF(). Recently, there has been an increased interest in studying -analogs of combinatorial designs from an applications view. These -analog structures can be useful in network coding and distributed storage, see e.g. [GPe8]. It is therefore natural to ask which combinatorial structures can be generalized from sets to vector spaces over GF(). For combinatorial designs, this uestion was first studied by Ray-Chaudhuri [BRC74], Cameron [Cam74a, Cam74b] and Delsarte [Del76] in the early 970s. Specifically, let GF() v be a vector space of dimension v over the finite field GF(). Then a t-(v, k, λ) subspace design is defined as a collection of Dipartimento di Matematica e Informatica, Universita degli Studi di Perugia, 0623 Perugia, Italy, buratti@dmi.unipg.it Department of Mathematics, University of Bayreuth, D-95440 Bayreuth, Germany, michael.kiermaier@uni-bayreuth.de Department of Mathematics, University of Bayreuth, D-95440 Bayreuth, Germany, sascha.kurz@uni-bayreuth.de Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 0000 Zagreb, Croatia, anamari.nakic@fer.hr Department of Mathematics, University of Bayreuth, D-95440 Bayreuth, Germany, alfred.wassermann@uni-bayreuth.de

k-dimensional subspaces of GF() v, called blocks, such that each t-dimensional subspace of GF() v is contained in exactly λ blocks. Such t-designs over GF() are the -analogs of conventional designs. By analogy with the case, a t-(v, k, ) subspace design is said to be a -Steiner system, and denoted S(t, k, v). Another well-known class of objects in combinatorial design theory are group divisible designs [MG07]. Considering the above, it therefore seems natural to ask for -analogs of group divisible designs. At first glance, this seems like a somewhat artificial task without much justification. But uite surprisingly, it turns out that -analogs of group divisible designs have interesting connections to scattered subspaces which are central objects in finite geometry, as well as to coding theory via r -divisible projective sets. We will also discuss the connection to -Steiner systems [BEÖ+ 6] and to design packings [EZ8]. Let K and G be sets of positive integers and let λ be a positive integer. A (v, K, λ, G)-group divisible design of index λ and order v is a triple (, G, B), where is a finite set of cardinality v, G, where #G >, is a partition of into parts (groups) whose sizes lie in G, and B is a family of subsets (blocks) of (with #B K for B B) such that every pair of distinct elements of occurs in exactly λ blocks or one group, but not both. See for example [MG07, Han75] for details. We note that the groups in group divisible designs have nothing to do with group theory. The -analog of a combinatorial structure over sets is defined by replacing subsets by subspaces and cardinalities by dimensions. Thus, the -analog of a group divisible design can be defined as follows. Definition Let K and G be sets of positive integers and let λ be a positive integer. A -analog of a group divisible design of index λ and order v denoted as (v, K, λ, G) -GDD is a triple (, G, B), where is a vector space over GF() of dimension v, G is a vector space partition of into subspaces (groups) whose dimensions lie in G, and B is a family of subspaces (blocks) of, that satisfies. #G >, 2. if B B then dim B K, 3. every 2-dimensional subspace of occurs in exactly λ blocks or one group, but not both. A (v, K, λ, {g}) -GDD is called g-uniform. Subseuently, if K or G are oneelement sets, we denote it by small letters, e.g. (v, k, λ, g) -GDD for K = {k} and G = {g}. A set of subspaces of such that every -dimensional subspace is covered exactly once is called vector space partition. 2

In the rest of the paper we study the case K = {k} and G = {g}. The latter implies that the vector space partition G is a partition of the -dimensional subspaces of in subspaces of dimension g. In finite geometry such a structure is known as (g )-spread. Additionally, we will only consider so called simple group divisible designs, i.e. designs without multiple appearances of blocks. A possible generalization would be to reuire the last condition in Definition for every t-dimensional subspace of, where t 2. For t = such a definition would make no sense. An euivalent formulation of the last condition in Definition would be that every block in B intersects the spread elements in dimension of at most one. The -analog of concept of a transversal design would be that every block in B intersects the spread elements exactly in dimension one. But for -analogs this is only possible in the trivial case g =, k = v. However, a related concept was defined in [ES3]. Among all 2-subspaces of, only a small fraction is covered by the elements of G. Thus, a (v, k, λ, g) -GDD is almost a 2-(v, k, λ) subspace design, in the sense that the vast majority of the 2-subspaces is covered by λ elements of B. From a slightly different point of view, a (v, k, λ, g) -GDD is a 2-(v, k, λ, g) packing design of fairly large size, which are designs where the condition each t-subspace is covered by exactly λ blocks is relaxed to each t-subspace is covered by at most λ blocks [BKW8a]. In Section 6 we give an example of a (6, 3, 2, 2) 2 -GDD consisting of 80 blocks. This is the largest known 2-(6, 3, 2) 2 packing design. We note that a -analog of a group divisible design can be also seen as a special graph decomposition over a finite field, a concept recently introduced in [BNW]. It is indeed euivalent to a decomposition of a complete m-partite graph into cliues where: the vertices are the points of a projective space PG(n, ); the parts are the members of a spread of PG(n, ) into subspaces of a suitable dimension; the vertex-set of each cliue is a subspace of PG(n, ) of a suitable dimension. 2 Preliminaries For m v we denote the set of m-dimensional subspaces of, also called Grassmannian, by [ ] m. It is well known that its cardinality can be expressed by the Gaussian coefficient [ ] [ ] v # = = (v )( v ) ( v m+ ) m m ( m )( m. ) ( ) Definition 2 Given a spread in dimension v, let [ k] be the set of all blocks that contain no 2-dimensional subspace which is already covered by the spread. The intersection between a k-dimensional subspace B [ 2] and all elements of the spread is at most one-dimensional. In finite geometry such a subspace B [ k] is called scattered subspace with respect to G [BBL00, BL00]. In case g =, i.e. G = [ ], no 2-dimensional subspace is covered by this trivial spread. Then, (, B) is a 2-(v, k, λ) subspace design. See [BKW8b, 3

BKW8a] for surveys about subspace designs and computer methods for their construction. Let g s = v and = GF() v. Then, the set of -dimensional subspaces of GF( g ) s regarded as g-dimensional subspaces in the -linear vector space GF() v, i.e. [ GF( g ) s ] G = is called Desarguesian spread. A t-spread G is called normal or geometric, if U, G then any element W G is either disjoint to the subspace U, or contained in it, see e.g. [Lun99]. Since all normal spreads are isomorphic to the Desarguesian spread [Lun99], we will follow [Lav6] and denote normal spreads as Desarguesian spreads. If s {, 2}, then all spreads are normal and therefore Desarguesian. The automorphism group of a Desarguesian spread G is PΓL(s, g ). Trivial -analogs of group divisible designs. For subspace designs, the empty set as well as the the set of all k-dimensional subspaces in GF() v always are a designs, called trivial designs. Here, it turns out that the uestion if trivial -analogs of group divisible designs exist is rather non-trivial. Of course, there exists always the trivial (v, k, 0, g) -GDD (, G, {}). But it is not clear if the set of all scattered k-dimensional subspaces, i.e. (, G, [ k] ), is always a -GDD. This would reuire that every subspace L [ ] 2 that is not covered by the spread, is contained in the same number λ max of blocks of [ k]. If this is the case, we call (, [ k], G) the complete (v, k, λ max, g) -GDD. If the complete (v, k, λ max, g) -GDD exists, then for any (v, k, λ, g) -GDD (, G, B) the triple (, G, [ 2] \ B) is a (v, k, λ max λ, g) -GDD, called the supplementary -GDD. For a few cases we can answer the uestion if the complete -GDD exists, or in other words, if there is a λ max. In general, the answer depends on the choice of the spread. In the smallest case, k = 3, however, λ max exists for all spreads. Lemma Let G be a (g )-spread in and let L be a 2-dimensional subspace which is not contained in any element of G. Then, L is contained in [ ] [ ] [ ] v 2 2 g λ max = 3 2 3 2 blocks of [ 3]. Proof. Every 2-dimensional subspace L is contained in [ ] v 2 3 2 3-dimensional subspaces of [ ] 3. If L is not contained in any spread element, this means that L intersects [ ] 2 different spread elements and the intersections are -dimensional. Let S be one such spread element. Now, there are [ ] g choices among the 3-dimensional subspaces in [ ] 3 which contain L to intersect S in dimension two. Therefore, L is contained in [ ] [ ] [ ] v 2 2 g λ max = 3 2 3 2 4 g,

blocks of [ 3]. In general, the existence of λ max may depend on the spread. This can be seen from the fact that the maximum dimension of a scattered subspace depends on the spread, see [BL00]. However, for a Desarguesian spread and g = 2, k = 4, we can determine λ max. Lemma 2 Let G be a Desarguesian (2 )-spread in and let L be a 2- dimensional subspace which is not contained in any element of G. Then, L is contained in [ ] [ ] [ ] [ ] [ ] [ ] [ ] v 2 2 v 4 v 2 4 2 λ max = / + / 4 2 blocks of [ 4]. Proof. Every 2-dimensional subspace L is contained in [ ] v 2 4 2 4-dimensional subspaces. If L is not covered by the spread this means that L intersects [ ] 2 spread elements S,..., S +, which span a 4-dimensional space F. All other spread elements are disjoint to L. Since L F, we have to subtract one possibility. For each i +, S i, L is contained in [ ] v 4 4-dimensional subspaces with a 3-dimensional intersection with F. All other spread elements S of F satisfy S, L = F. If S is one of the [ v ] /[ 2 ] [ 4 ] /[ ] 2 spread elements disjoint from F, then F := S, L intersects F in dimension 2. Moreover, F does not contain any further spread element, since otherwise F would be partitioned into 2 + spread elements, where + of them have to intersect L. Thus, L is contained in exactly λ max elements from [ 4]. 3 Necessary conditions on (v, k, λ, g) The necessary conditions for a (v, k, λ, g)-gdd over sets are g v, k v/g, λ( v g )g 0 (mod k ), and λ v g ( v g )g2 0 (mod k(k )), see [Han75]. For -analogs of GDDs it is well known that (g )-spreads exist if and only if g divides v. A (g )-spread consists of [ v ] /[ ] g blocks and contains [ ] g 2 [ ] v / [ ] g 2-dimensional subspaces. Based on the pigeonhole principle we can argue that if B is a block of a (v, k, λ, g) -GDD then there can not be more points in B than the number of spread elements, i.e. if [ k ] [ v ] /[ ] g. It follows that (see [BL00, Theorem 3.]) k v g. () This is the -analog of the restriction k v/g for the set case. If G is a Desarguesian spread, it follows from [BL00, Theorem 4.3] for the parameters (v, k, λ, g) to be admissible that k v/2. 5

By looking at the numbers of 2-dimensional subspaces which are covered by spread elements we can conclude that the cardinality of B has to be [ v 2] #B = λ [ g 2 ] [v ] /[ ] g [ k. (2) 2] A necessary condition on the parameters of a g-uniform -(k, λ) GDD is that the cardinality in (2) is an integer number. Any fixed -dimensional subspace P is contained in [ ] v 2-dimensional subspaces. Further, P lies in exactly one block of the spread and this block covers [ ] g 2-dimensional subspaces through P. Those 2-dimensional subspaces are not covered by blocks in B. All other 2-dimensional subspaces containing P are covered by exactly λ k-dimensional blocks. Such a block contains P and there are [ ] k 2-dimensional subspaces through P in this block. It follows that P is contained in exactly [ v ] λ [ ] g [ k ] (3) k-dimensional blocks and this number must be an integer. The number (3) is the replication number of the point P in the -GDD. Up to now, the restrictions (), (2), (3), as well as g divides v, on the parameters of a (v, k, λ, g) -GDD are the -analogs of restrictions for the set case. But for -GDDs there is a further necessary condition whose analog in the set case is trivial. Given a multiset of subspaces of, we obtain a corresponding multiset P of points by replacing each subspace by its set of points. A multiset P [ ] of points in can be expressed by its weight function w P : For each point P we denote its multiplicity in P by w P (P ). We write #P = P w P (P ) and #(P H) = P H w P (P ) where H is an arbitrary hyperplane in. Let r < v be an integer. If #P #(P H) (mod r ) for every hyperplane H, then P is called r -divisible. 2 In [KK7, Lemma ] it is shown that the multiset P of points corresponding to a multiset of subspaces with dimension at least k is k -divisible. Lemma 3 ([KK7, Lemma ]) For a non-empty multiset of subspaces of with m i subspaces of dimension i let P be the corresponding multiset of points. If m i = 0 for all 0 i < k, where k 2, then for every hyperplane H. #P #(P H) (mod k ) 2 Taking the elements of P as columns of a generator matrix gives a linear code of length #P and dimension k whose codewords have weights being divisible by r. 6

Proof. We have #P = v i=0 m v ] i[ i. The intersection of an i-subspace U with an arbitrary hyperplane H has either dimension i or i. Therefore, for the set P of points corresponding to U, we get that #P = [ ] i and that #(P H) is eual to [ i ] or [ i ]. In either case, it follows from [ i ] [ ] i (mod i ) that [ ] i #(P H) (mod i ). Summing up yields the proposed result. If there is a suitable integer λ such that w P (P ) λ for all P, then we can define for P the complementary weight function w λ (P ) = λ w(p ) which in turn gives rise to the complementary multiset of points P. In [KK7, Lemma 2] it is shown that a r -divisible multiset P leads to a multiset P that is also r -divisible. Lemma 4 ([KK7, Lemma 2]) If a multiset P in is r -divisible with r < v and satisfies w P (P ) λ for all P then the complementary multiset P is also r -divisible. Proof. We have [ ] [ ] # P v = λ #P and #( P v H) = λ #(P H) for every hyperplane H. (mod r ) which holds for r < v. Thus, the result follows from [ v ] [ v These easy but rather generally applicable facts about r -divisible multiset of points are enough to conclude: Lemma 5 Let (, G, B) be a (v, k, λ, g) -GDD and 2 g k, then k g divides λ. Proof. Let P [ ] be an arbitrary point. Then there exists exactly one spread element S G that contains P. By B P we denote the elements of B that contain P. Let S and B P denote the corresponding subspaces in the factor space /P. We observe that every point of [ ] S is disjoint to the elements of B P and that every point in [ /P ] \ [ ] S is met by exactly λ elements of B P (all having dimension k ). We note that B P gives rise to a k 2 -divisible multiset P of points. So, its complement P, which is the λ-fold copy of S, also has to be k 2 -divisible. For every hyperplane H not containing S, we have #( P H) = λ [ g 2 ] and # P = λ [ ] g. Thus, λg 2 = # P #( P H) 0 (mod k 2 ), so that k g divides λ. We remark that the criterion in Lemma 5 is independent of the dimension v of the ambient space. Summarizing the above we arrive at the following restrictions. ] 7

Theorem Necessary conditions for a (v, k, λ, g) -GDD are. g divides v, 2. k v g, 3. the cardinalities in (2), (3) are integer numbers, 4. if 2 g k then k g divides λ. If these conditions are fulfilled, the parameters (v, k, λ, g) are called admissible. Table contains the admissible parameters for = 2 up to dimension v = 4. Column λ gives the minimum value of λ which fulfills the above necessary conditions. All admissible values of λ are integer multiples of λ. In column #B the cardinality of B is given for λ = λ. Those values of λ max that are valid for the Desarguesian spread only are given in italics, where the values for (v, g, k) = (8, 4, 4) and (9, 3, 4) have been checked by a computer enumeration. For the case λ =, the online tables [HKKW6] http://subspacecodes.uni-bayreuth.de may give further restrictions, since B is a constant dimension subspace code of minimum distance 2(k ) and therefore #B A (v, 2(k ); k). The currently best known upper bounds for A (v, d; k) are given by [HHK + 7, Euation (2)] referring back to partial spreads and A 2 (6, 4; 3) = 77 [HKK5], A 2 (8, 6; 4) = 257 [HHK + 7] both obtained by exhaustive integer linear programming computations, see also [KK7]. 4 -GDDs and -Steiner systems In the set case the connection between Steiner systems 2-(v, k, ) and group divisible designs is well understood. Theorem 2 ([Han75, Lemma 2.2]) A 2-(v +, k, ) design exists if and only if a (v, k,, k )-GDD exists. There is a partial -analog of Theorem 2: Theorem 3 If there exists a 2-(v+, k, ) subspace design, then a (v, k, 2, k ) -GDD exists. Proof. Let be a vector space of dimension v + over GF(). We fix a point P [ ] and define the projection π : PG( ) PG( /P ), U (U + P )/P. For any subspace U we have { dim(u) if P U, dim(π(u)) = dim(u) otherwise. 8

Table : Admissible parameters for (v, k, λ, g) 2 -GDDs with v 4. v g k λ λ max #B #G 6 2 3 2 2 80 2 6 3 3 3 6 252 9 8 2 3 2 60 3060 85 8 2 4 4 480 224 85 8 4 3 7 42 0200 7 8 4 4 7 4 2040 7 9 3 3 8 632 73 9 3 4 0 680 2264 73 0 2 3 4 252 347820 34 0 2 4 28 0080 3928 34 0 2 5 8 8976 34 0 5 3 2 20 507408 33 0 5 4 35 6936 33 0 5 5 5 6368 33 2 2 3 2 020 797940 365 2 2 4 28 7360 2234232 365 2 2 5 40 720720 365 2 2 6 6 68640 365 2 3 3 3 04 95740 585 2 3 4 2 59432 585 2 3 5 860 33480720 585 2 3 6 248 062880 585 2 4 3 002 397800 273 2 4 4 7 556920 273 2 4 5 62 3840 273 2 4 6 24 530400 273 2 6 3 930 39320 65 2 6 4 78624 65 2 6 5 55 275840 65 2 6 6 3 3040 65 4 2 3 2 4092 2778740 546 4 2 4 4 2782560 5496 546 4 2 5 248 7560944 546 4 2 6 496 34076640 546 4 2 7 32 536640 546 4 7 3 2 3906 336024 29 4 7 4 35 44387008 29 4 7 5 465 336024 29 4 7 6 65 44387008 29 4 7 7 63 04852 29 9

Let D = (, B ) be a 2-(v +, k, ) subspace design. The set G = {π(b) B B, P B} is the derived design of D with respect to P [KL5], which has the parameters -(v, k, ). In other words, it is a (k 2)-spread in /P. Now define B = {π(b) B B, P / B} and = /P. We claim that (, G, B) is a (v, k, 2, k ) -GDD. In order to prove this, let L [ ] 2 be a line not covered by any element in G. Then L = E/P, where E [ ] 3, P E and E is not contained in a block of the design D. The blocks of B covering L have the form π(b) with B B such that B E is a line in E not passing through P. There are 2 such lines and each line is contained in a uniue block in B. Since these 2 blocks B have to be pairwise distinct and do not contain the point P, we get that there are 2 blocks π(b) B containing L. Since there are 2-(3, 3, ) 2 subspace designs [BEÖ+ 6], by Theorem 3 there are also (2, 3, 4, 2) 2 -GDDs. The smallest admissible case of a 2-(v, 3, ) subspace design is v = 7, which is known as a -analog of the Fano plane. Its existence is a notorious open uestion for any value of. By Theorem 3, the existence would imply the existence of a (6, 3, 2, 2) -GDD, which has been shown to be true in [EH7] for any value of, in the terminology of a residual construction for the - Fano plane. In Theorem 4, we will give a general construction of -GDDs covering these parameters. The crucial uestion is if a (6, 3, 2, 2) -GDD can be lifted to a 2-(7, 3, ) subspace design. While the GDDs with these parameters constructed in Theorem 4 have a large automorphism group, for the binary case = 2 we know from [BKN6, KKW8] that the order of the automorphism group of a putative 2-(7, 3, ) 2 subspace design is at most two. So if the lifting construction is at all possible for the binary (6, 3, 4, 2) 2 -GDD from Theorem 4, necessarily many automorphisms have to get destroyed. In Table 2 we can see that there exists a (8, 3, 4, 2) 2 -GDD. This might lead in the same way to a 2-(9, 3, ) 2 subspace design, which is not known to exist. 5 A general construction A very successful approach to construct t-(v, k, λ) designs over sets is to prescribe an automorphism group which acts transitively on the subsets of cardinality t. However for -analogs of designs with t 2 this approach yields only trivial designs, since in [CK79, Prop. 8.4] it is shown that if a group G PΓL(v, ) acts transitively on the t-dimensional subspaces of, 2 t v 2, then G acts transitively also on the k-dimensional subspaces of for all k v. The following lemma provides the counterpart of the construction idea for -analogs of group divisible designs. Unlike the situation of -analogs of designs, in this slightly different setting there are indeed suitable groups admitting the general construction of non-trivial -GDDs, which will be described in the seuel. Itoh s construction of infinite families of subspace designs is based on a similar idea [Ito98]. 0

Lemma 6 Let G be a (g )-spread in PG( ) and let G be a subgroup of the stabilizer PΓL(v, ) G of G in PΓL(v, ). If the action of G on [ ] 2 \ [ S S G 2] is transitive, then any union B of G-orbits on the set of k-subspaces which are scattered with respect to G yields a (v, k, λ, g) -GDD (, G, B) for a suitable value λ. Proof. By transitivity, the number λ of blocks in B passing through a line L [ ] 2 \ [ S S G 2] does not depend on the choice of L. In the following, let = GF( g ) s, which is a vector space over GF() of dimension v = gs. Furthermore, let G = [ ] be the Desarguesian (g )- g spread in PG( ). For every GF()-subspace U we have that dim GF(g )( U GF(g )) dimgf() (U). In the case of euality, U will be called fat. Euivalently, U is fat if and only if one (and then any) GF()-basis of U is GF( g )-linearly independent. The set of fat k-subspaces of will be denoted by F k. We remark that for a fat subspace U, the set of points { x GF(g ) : x U} is a Baer subspace of as a GF( g )-vector space. Lemma 7 #F k = (g )(k 2) k i=0 g(s i) k i. Proof. A seuence of k vectors in is the GF()-basis of a fat k-subspace if and only if it is linearly independent over GF( g ). Counting the set of those seuences in two ways yields k #F k ( k i ) = i=0 k (( g ) s ( g ) i ), i=0 which leads to the stated formula. We will identify the unit group GF() with the corresponding group of s s scalar matrices over GF( g ). Lemma 8 Consider the action of SL(s, g )/ GF() on the set of the fat k- subspaces of. For k < s, the action is transitive. For k = s, F k splits into orbits of eual length. g Proof. Let U be a fat k-subspace of and let B be an ordered GF()-basis of U. Then B is an ordered GF( g )-basis of U GF( g ). For k < s, B can be extended to an ordered GF( g )-basis B of. Let A be the (s s)-matrix over GF( g ) whose columns are given by B. By scaling one of the vectors in B \ B, we may assume det(a) =. Now the mapping, x Ax is in SL(s, g ) and maps the fat k-subspace e,..., e k to U (e i denoting the i-th standard vector of ). Thus, the action of SL(s, g )/ GF() is transitive on F k. It remains to consider the case k = s. Let A be the (s s)-matrix over GF( g ) whose columns are given by B. As any two GF()-bases of U can be

mapped to each other by a GF()-linear map, we see that up to a factor in GF(), det(a) does not depend on the choice of B. Thus, det(u) := det(a) GF() GF( g ) / GF() is invariant under the action of SL(s, g ) on F k. It is readily checked that every value in GF( g ) / GF() appears as the invariant det(u) for some fat s-subspace U, and that two fat s-subspaces having the same invariant can be mapped to each other within SL(s, g ). Thus, the number of orbits of the action of SL(s, g ) on F s is given by the number #(GF( g ) / GF() ) = g of invariants. As SL(s, g ) is normal in GL(s, g ) which acts transitively on F s, all orbits have the same size. Modding out the kernel GF() of the action yields the statement in the lemma. Theorem 4 Let be a vector space over GF() of dimension gs with g 2 and s 3. Let G be a Desarguesian (g )-spread in PG( ). For k {3,..., s }, (, G, F k ) is a (gs, k, λ, g) -GDD with k λ = (g )((k 2) ) i=2 g(s i) k i. Moreover, for each α {,..., g }, the union B of any α orbits of the action of SL(s, g )/ GF() on F s gives a (gs, s, λ, g) -GDD (, G, B) with s 2 λ = α (g )((s 2) ) i=2 gi i. Proof. We may assume = GF( g ) s and G = [ ]. The lines covered by g the elements of G are exactly the non-fat GF()-subspaces of of dimension 2. By Lemma 6 and Lemma 8, (, G, F k ) is a GDD. Double counting yields #F 2 λ = #F k [k]. Using Lemma 7, this euation transforms into the given 2 formula for λ. In the case k = s, by Lemma 8, each union B of α {,..., g } orbits under the action of SL(s, )/ GF() on F s yields a GDD with s λ = α (g )((s 2) ) g(s i) g s i = 2) ) α(g )((s i=2 s 2 i=2 gi i. Remark In the special case g = 2, s = 3 and α = the second case of Theorem 4 yields (6, 3, 2, 2) -GDDs. These parameters match the residual construction for the -Fano plane in [EH7]. Remark 2 A fat k-subspace (k {3,..., s}) is always scattered with respect to the Desarguesian spread [ ]. The converse is only true for g = 2. Thus, g Theorem 4 implies that the set of all scattered k-subspaces with respect to the Desarguesian line spread of GF() 2s is a (2s, k, λ max, 2) -GDD. 2

Table 2: Existence results for (v, k, λ, g) -GDD for = 2. v g k λ λ max λ comments 6 2 3 2 2 4 [EH7] 2, 4,..., 2 σ 7 4α, α =, 2, 3 Thm. 4 6 3 3 3 6 3, 6 σ 2 8 2 3 2 60 2, 58 σ, φ 4 4, 6,..., 54, 56, 60 σ, φ 8 2 4 4 480 20, 40,..., 480 σ, φ 60α, α =, 2, 3 Thm. 4 8 4 3 7 42 7, 2, 35 σ 4, 28, 42 σ, φ 8 4 4 7 4 7, 4 σ 9 3 3 8 2, 3,..., 5, 6, 8 σ, φ 6α, α =,..., 6 Thm. 4 9 3 4 0 680 30, 60,..., 680 σ, φ 0 2 3 4 252 4, 28,..., 252 σ, φ 0 2 5 8 23040α, α =,..., 3 Thm. 4 0 5 3 2 20 05, 20 σ, φ 2 2 2 3 2 020 4 [BEÖ+ 6] 2 2 6 6 2533760α, α =,..., 3 Thm. 4 2 3 4 2 2504α, α =,..., 7 Thm. 4 2 4 3 002 64α, α =,..., 5 Thm. 4 6 Computer constructions An element π PΓL(v, ) is an automorphism of a (v, k, λ, g) -GDD if π(g) = G and π(b) = B. Taking the Desarguesian (g )-spread and applying the Kramer-Mesner method [KM76] with the tools described in [BKL05, BKW8b, BKW8a] to the remaining blocks, we have found (v, k, λ, g) -GDDs for the parameters listed in Tables 2, 3. In all cases, the prescribed automorphism groups are subgroups of the normalizer σ, φ of a Singer cycle group generated by an element σ of order v and by the Frobenius automorphism φ, see [BKW8a]. Note that the presented necessary conditions for λ turn out to be tight in several cases. Example. We take the primitive polynomial + x + x 3 + x 4 + x 6, together with the canonical Singer cycle group generated by 00000 00000 σ = 00000 00000 00000 00 For a compact representation we will write all α β matrices X over GF() 3

Table 3: Existence results for (v, k, λ, g) -GDD for = 3. v g k λ λ max λ comments 6 2 3 3 36 9 [EH7] 9α, α =,..., 4 Thm. 4 2, 8 σ 3, φ 6 3 3 4 24 2, 24 σ 4, φ 8 2 4 9 9720 2430α, α =,..., 4 Thm. 4 8 4 3 3 32 52, 04, 56, 208, 260, 32 σ, φ 9 3 3 077 8α, α =,..., 3 Thm. 4 0 2 5 27 22044960 55240α, α =,..., 4 Thm. 4 2 2 6 8 439267872960 0986968240α, α =,..., 4 Thm. 4 2 3 4 3 5373459α, α =,..., 3 Thm. 4 2 4 3 29472 729α, α =,..., 40 Thm. 4 with entries x i,j, whose indices are numbered from 0, as vectors of integers [ j x 0,j j,..., j x α,j j ], i.e. σ = [2, 4, 8, 6, 32, 27]. The block representatives of a (6, 3, 2, 2) 2 -GDD can be constructed by prescribing the subgroup G = σ 7 of the Singer cycle group. The order of G is 9, a generator is [54, 55, 53, 49, 57, 4]. The spread is generated by [, 4], under the action of G the 2 spread elements are partitioned into 7 orbits. The blocks of the GDD consist of the G-orbits of the following 20 generators. [3, 6, 32], [5, 6, 32], [4, 8, 32], [5, 8, 32], [9, 24, 32], [7, 24, 32], [0, 4, 32], [8, 28, 32], [7, 20, 32], [, 28, 32], [7, 0, 32], [25, 2, 32], [3, 6, 32], [29, 30, 32], [33, 2, 6], [38, 40, 6], [2, 36, 6], [, 36, 6], [, 2, 6], [9, 20, 8] Acknowledgements The authors are grateful to Anton Betten who pointed out the connection to scattered subspaces. References [BBL00] Simeon Ball, Aart Blokhuis, and Michel Lavrauw. Linear (+)-fold blocking sets in PG(2, 4 ). Finite Fields and Their Applications, 6(4):294 30, 2000. [BEÖ+ 6] Michael Braun, Tuvi Etzion, Patric R. J. Östergård, Alexander ardy, and Alfred Wassermann. Existence of -analogs of Steiner systems. Forum of Mathematics, Pi, 4:e7 (4 pages), 206. 4

[BKL05] Michael Braun, Adalbert Kerber, and Reinhard Laue. Systematic construction of -analogs of t-(v, k, λ)-designs. Designs, Codes and Cryptography, 34():55 70, 2005. [BKN6] Michael Braun, Michael Kiermaier, and Anamari Nakić. On the automorphism group of a binary -analog of the Fano plane. European Journal of Combinatorics, 5:443 457, 206. [BKW8a] Michael Braun, Michael Kiermaier, and Alfred Wassermann. Computational methods in subspace designs. In Network Coding and Subspace Designs, Signals and Communication Technology, pages 23 244. Springer, Cham, 208. [BKW8b] Michael Braun, Michael Kiermaier, and Alfred Wassermann. - analogs of designs: Subspace designs. In Network Coding and Subspace Designs, Signals and Communication Technology, pages 7 2. Springer, Cham, 208. [BL00] Aart Blokhuis and Michel Lavrauw. Scattered spaces with respect to a spread in PG(n, ). Geometriae Dedicata, 8():23 243, Jul 2000. [BNW] Marco Buratti, Anamari Nakić, and Alfred Wassermann. Graph decompositions in projective geometries. in preparation. [BRC74] Claude Berge and Dijen Ray-Chaudhuri. Unsolved problems. In Claude Berge and Dijen Ray-Chaudhuri, editors, Hypergraph Seminar: Ohio State University 972, number 4 in Lecture Notes in Mathematics, pages 278 287. Springer, Berlin, Heidelberg, 974. [Cam74a] Peter J. Cameron. Generalisation of Fisher s ineuality to fields with more than one element. In T. P. McDonough and. C. Mavron, editors, Combinatorics - Proceedings of the British Combinatorial Conference 973, number 3 in London Mathematical Society Lecture Note Series, pages 9 3. Cambridge University Press, Cambridge, 974. [Cam74b] Peter J. Cameron. Locally symmetric designs. Geometriae Dedicata, 3:65 76, 974. [CK79] Peter J. Cameron and William M. Kantor. 2-transitive and antiflag transitive collineation groups of finite projective spaces. Journal of Algebra, 60(2):384 422, 979. [Del76] Philippe Delsarte. Association schemes and t-designs in regular semilattices. Journal of Combinatorial Theory, Series A, 20(2):230 243, 976. [EH7] Tuvi Etzion and Niv Hooker. Residual -Fano planes and related structures. ArXiv e-prints, April 207. [ES3] Tuvi Etzion and Natalia Silberstein. Codes and designs related to lifted MRD codes. IEEE Transactions on Information Theory, 59(2):004 07, 203. 5

[Eul53] Leonhard Euler. Consideratio uarumdam serierum uae singularibus proprietatibus sunt praeditae. Novi Commentarii academiae scientiarum imperialis Petropolitanae, III:86 08, 750 & 75/753. [EZ8] Tuvi Etzion and Hui Zhang. Grassmannian Codes with New Distance Measures for Network Coding. ArXiv e-prints, January 208. [GPe8] Marcus Greferath, Mario Osvin Pavčević, and Natalia Silberstein María Ángeles áuez-castro (eds.). Network Coding and Subspace Designs. Signals and Communication Technology. Springer, Cham, 208. [Han75] Haim Hanani. Balanced incomplete block designs and related designs. Discrete Mathematics, (3):255 369, 975. [HHK + 7] Daniel Heinlein, Thomas Honold, Michael Kiermaier, Sascha Kurz, and Alfred Wassermann. Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6. ArXiv e-prints, November 207. [HKK5] Thomas Honold, Michael Kiermaier, and Sascha Kurz. Optimal binary subspace codes of length 6, constant dimension 3 and minimum subspace distance 4. Contemporary Mathematics, 632:57 76, 205. [HKKW6] Daniel Heinlein, Michael Kiermaier, Sascha Kurz, and Alfred Wassermann. Tables of subspace codes. Technical report, Universität Bayreuth, Bayreuth, Januar 206. https://epub.unibayreuth.de/2670/. [Ito98] Toyoharu Itoh. A new family of 2-designs over GF() admitting SL m ( l ). Geometriae Dedicata, 69(3):26 286, 998. [KK7] Michael Kiermaier and Sascha Kurz. An improvement of the Johnson bound for subspace codes. ArXiv e-prints, July 207. [KKW8] Michael Kiermaier, Sascha Kurz, and Alfred Wassermann. The order of the automorphism group of a binary -analog of the Fano plane is at most two. Designs, Codes and Cryptography, 86(2):239 250, Feb 208. [KL5] Michael Kiermaier and Reinhard Laue. Derived and residual subspace designs. Advances in Mathematics of Communications, 9():05 5, 205. [KM76] Earl S. Kramer and Dale M. Mesner. t-designs on hypergraphs. Discrete Mathematics, 5:263 296, 976. [Lav6] Michel Lavrauw. Scattered spaces in Galois geometry. In Anne Canteaut, Gove Effinger, Sophie Huczynska, Daniel Panario, and Leo Storme, editors, Contemporary developments in finite fields and applications, pages 95 26. World Sci. Publ., Hackensack, NJ, 206. 6

[Lun99] Guglielmo Lunardon. Normal spreads. Geometriae Dedicata, 75(3):245 26, May 999. [MG07] Ronald C. Mullin and Hans-Dietrich O. F. Gronau. PBDs and GDDs: The basics. In Charles J. Colbourn and Jeffrey H. Dinitz, editors, Handbook of Combinatorial Designs, chapter I., pages 23 236. Chapman & Hall/CRC, 2 edition, 2007. [Tit57] Jacues L. Tits. Sur les analogues algébriues des groupes semisimples complexes. In Colloue d algèbre supérieure, tenu à Bruxelles du 9 au 22 décembre 956, Centre Belge de Recherches Mathématiues, pages 26 289. Établissements Ceuterick, Louvain; Librairie Gauthier-illars, Paris, 957. 7