Some new large sets of geometric designs of type LS [ 3 ] [ 2 , 3 , 2 8 ] Research

Let V be an n-dimensional vector space over Fq. By a geometric t-[q, k, λ] design we mean a collection D of k-dimensional subspaces of V , called blocks, such that every t-dimensional subspace T of V appears in exactly λ blocks in D. A large set, LS[N][t, k, q], of geometric designs, is a collection of N t-[q, k, λ] designs which partitions the collection [ V k ] of all k-dimensional subspaces of V . Prior to recent article [4] only large sets of geometric 1-designs were known to exist. However in [4] M. Braun, A. Kohnert, P. Östergard, and A. Wasserman constructed the world’s first large set of geometric 2-designs, namely an LS[3][2,3,2], invariant under a Singer subgroup in GL8(2). In this work we construct an additional 9 distinct, large sets LS[3][2,3,2], with the help of lattice basis-reduction. 2010 MSC: 05B25, 05B40, 05E18


Introduction
In this article we deal with large sets of geometric t-designs.By a geometric t-design we mean what earlier authors have called t-designs over a finite field, or designs on vector spaces.Geometric t-designs are the F q -analogs of ordinary t-(v, k, λ) designs.The earliest mention of t-[q n , k, λ] designs, although not using our terminology or notation, was by P.J. Cameron in 1974 [5,6] and P. Delsarte in 1976 [7].In 1987, S. Thomas [20] exhibited the first simple geometric 2-design, and in the 1990's H. Suzuki [19], M. Miyakawa et al. [17] , and T. Itoh [10] constructed new geometric 2-designs and families of such designs.In 1994, D.K. Ray-Chaudhuri and E.J. Schram [18] studied and constructed geometric t-designs from quadratic forms, allowing repeated blocks.For the first time, the latter authors also studied large sets of geometric t-designs.
In a short recent arXiv preprint [8], and based on a probabilistic existence theorem of G. Kuperberg, S. Lovett and R. Peled in preprint [14], A. Fazeli, S. Lovett, and A. Vardy, appear to have proved the remarkable theorem that simple geometric t-designs exist for all values of t.This would be a q-analog of the famous theorem of L. Tierlinck for ordinary t-designs.It should be noted however, that the result in [8] is purely existential and there is no known efficient algorithm which can produce t-[q n , k, λ] designs for t > 3. The authors present the following challenge: Problem 1.1.Design an efficient algorithm to produce simple, non-trivial t-[q n , k, λ] designs for large t, (say t ≥ 4).
Of course, finding large sets of geometric t-designs is even harder than just finding geometric tdesigns.Prior to recent article [4] only large sets of geometric 1-designs were known to exist.However in [4] M. Braun, A. Kohnert, P. Östergard, and A. Wasserman constructed the world's first large set of geometric 2-designs, namely an LS [3][2,3,2 8 ], invariant under a Singer subgroup in GL 8 (2).
In this paper we construct 9 distinct large sets LS [3][2, 3, 2 8 ], all different from the large set constructed in [4].The computation involved our APL package knuth for group theoretic matters, and various LLL variants in the NTL library, augmented by certain optimization techniques for parallel lattice basis reduction.
It should be noted that some of the recent work on geometric t-designs has been motivated by present day coding theoretic applications as discussed in [9] and [11].

Preliminaries
Let V be an n-dimensional vector space over the field F q .If U is a j-dimensional subspace of V , we say that U is a j-subspace of V .If X is a set and 0 ≤ s ≤ |X|, X s denotes the collection of all subsets of cardinality s of X.
A geometric t-[q n , k, λ] design is a pair (V, B) where B is a multiset of k-subspaces of V , called blocks, such that any t-subspace T of V is contained in exactly λ blocks.(V, B) is said to be simple if B is a set, i.e. if there are no repeated blocks.
In this paper we deal only with simple geometric designs, and the square brackets of the symbol t-[q n , k, λ] will imply "geometric" in contrast to the round parentheses for an ordinary t-(v, k, λ) design.
We denote the collection of all k-subspaces of V by V k and note that | V k | = n k q , where n k q is the well known Gaussian binomial coefficient, given by: where for positive integer r, Analogously to the case of ordinary t-(v, k, λ) designs, a geometric t-[q n , k, λ] design (V, B) is also an s-[q n , k, λ i ] design for every 0 ≤ s ≤ t with: Thus, a necessary condition for the existence of a t-[q n , k, λ] design is that the λ s given by the equations (3) must be integral for all 0 ≤ s ≤ t.

By a large set LS
We can immediately see that for a given large set LS[N ][t, k, q n ], N can be expressed in terms of the other parameters as : Two t-[q n , k, λ] designs D = (V, B) and D = (V, B ) are said to be isomorphic if there exists α ∈ GL n (q) such that B α = B , that is, B α ∈ B for all B ∈ B, in which case we also write D α = D .If D α = D, then α is said to be an automorphism of D. The group of all automorphisms of D is denoted by Aut(D).
Equivalently, we say that a large set with this property is G-invariant.The group of all automorphisms of L is denoted by Aut(L).If the stronger condition holds, that B g i =B i for all B i ∈ B and g ∈ G, we say that the large set L is [G]-invariant.
In 1976, E.S. Kramer and D.M. Mesner [12] presented a theorem which provides necessary and sufficient conditions for the existence of an ordinary G-invariant t-(v, k, λ) design.Beginning with a given group action G|X, the authors define certain integer matrices, presently known as the Kramer-Mesner (KM) matrices.Roughly speaking such a matrix A t,k is the result of fusing under G the incidence matrix between X t and X k where incidence is set inclusion (fused R.Wilson matrix).These matrices extend naturally to the case of a group G ≤ GL n (q) acting on AG n (q) or P G n−1 (q), and provide necessary and sufficient conditions for the existence of geometric, G-invariant t-[q n , k, λ] designs.We proceed to define these matrices in the context of geometric t-designs, and state the analog of the Kramer-Mesner theorem.
Let V be an n-dimensional vector space over F q , and G ≤ GL n (q).Suppose that t and k are integers, 0 ≤ t < k ≤ n, and consider the actions of G on V t and V k respectively, with corresponding G-orbit decompositions: and where ρ(s) denotes the number of G-orbits on X s .Just as in [12], it can be shown that for any fixed t-subspaces T, T ∈ ∆ i , we have that that is, the number a t,k (i, j) = |{K ∈ Γ j : T ≤ K}| is independent of the choice of a fixed T ∈ ∆ i .The Kramer-Mesner matrix A t,k is then defined as the ρ(t) × ρ(k) matrix : Dually, for K fixed in Γ j , let b t,k (i, j) := |{T ∈ ∆ i : T ≤ K}|, and define the dual KM matrix B t,k by: In the following Lemma we state without proof geometric analogs of some properties of the A t,k and B t,k as included for the ordinary t-design context in [13].
Lemma 2.1.Let A t,k and B t,k , ∆ i , Γ j be as defined above.
Keeping in mind that we are only interested in simple geometric t-designs, we now state, without proof, the Kramer-Mesner theorem for geometric t-designs : Theorem 2.2.If G ≤ GL n (q), there is a G-invariant (simple) t-[q n , k, λ] design if and only if there is a ρ(k) × 1 0-1 vector u which is solution of the matrix equation where J is the ρ(t) × 1 vector of all 1's.
Here, the 1's in a solution u select the G-orbits of V k whose union will constitute the design.The following corollary follows immediately: of geometric designs if and only if there exist N distinct solutions, u 1 , . . ., u N , to the matrix equation (10), whose sum is the ρ(k) × 1 all 1's vector.

Main result
It is well known that GL n (q) has a cyclic subgroup of order q n − 1 , called a Singer subgroup, acting regularly on the non-zero vectors of V = F n q .It is also known that all Singer subgroups are conjugate in GL n (q).A Singer subgroup G of Γ = GL 8 (2) is the centralizer of a Sylow-17 subgroup of Γ and its normalizer N in Γ is a split extension of G by its Frobenius group Φ 8 , thus |N | = 2040.In particular, for the rest of the paper we adopt the notation V = F 8  2 , Γ = GL 8 (2), and G = α , where α is the same Singer cycle as the one used in [4], that is : 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 We will presently construct 9 distinct large sets of geometric 2 − [2 8 , 3, 21] designs which are [G]invariant under Singer subgroup G of Γ.We have used the exact same Singer subgroup G = α as in [4] so that it will be easy to check that our large sets are different from the one constructed in [4].

Members of V
2 are Klein 4-groups, and those of V 3 elementary abelian groups of order 8. Viewed projectively, the 2-and 3-spaces can be seen as collinear triples and Fano planes respectively.There are in all 10795 2-spaces, and 97155 3-spaces.
We begin by computing the G-orbits on V 2 and V 3 , where G = α .There are exactly 43 G-orbits on V 2 , all of which have length 255, except for one which has length 85.The short orbit is explained by the fact that the cyclic subgroup of order 3 in G fixes a collinear triple.There are 381 G-orbits on V 3 all of length 255.
The vectors of V = F 8 2 are represented by the radix-2 representation of integers in Z 256 .Orbits of 2and 3-spaces are represented by the lexically smallest basis among all members of the orbit, but since G is transitive on the non-zero vectors, each such basis will consist of the vector 1 ↔ 00000001, and one (or two) elements of Z 256 − {1}.Hence, to represent α -orbits of 2-spaces, it suffices to specify the second vector in the lexically minimal basis over all 2-spaces for that orbit.Thus, the α -orbits of 2-spaces are represented by the following 43 integers : Similarly, orbit representatives of the 381 orbits of 3-spaces are given by the pair of integers in Z 256 which together with 1, form the lexically minimal basis among the members of the orbit of 3-spaces.The pairs x, y ∈ Z 256 representing the G-orbits on 3-spaces will appear in our display of the KM-matrix A 2,3 below.
To compute A 2,3 , we found it easier to first compute matrix B 2,3 and then compute the A 2,3 (i, j) entries, using Lemma 2.1, equation (iii) : Almost all ratios , that is all, except for those involving the short orbit ∆43 of length 85, in which case the ratio is 3.For any particular Fano plane F in orbit Γj, it is easy to determine how the 7 lines of F are distributed among the orbits {∆i}, thus computation of B2,3 is straightforward.
In an effort to overcome the difficulty of presenting in this article the 43 × 381 matrix A2,3, the next two pages display a coded version of A2,3 from which, with a little effort, a user-friendly version of A2,3 can be recovered.Each column of A2,3 is a vector consisting of 43 elements from {0, 1, 3}.We adjoin two extra 0's at the top of the column and transpose, transforming the column to a row vector v ∈ Z 45 4 .We then use the following alphabet of 64 characters, as digits with values from 0 to 63: 0123456789abcdefghijklmnopqrstuvwxyzABCDEGFHIJKLMNOPQRSTUVWXYZ+− .

The vector v ∈ Z 45
4 is separated into 15 triples, and each triple, belonging to Z 3 4 , is encoded as a symbol in the alphabet using radix-4 notation.For example, 1000110101000101000000000000000000000000000 =⇒ 001 000 110 101 000 101 000 000 000 000 000 000 000 000 000 =⇒ 1 0 k h 0 h 0 0 0 0 0 0 0 0 0 =⇒ 10kh0h000000000 The next page displays the first 192 columns, and the subsequent page the remaining columns of A2,3.The 381 columns of A2,3 correspond to 381 short rows in the display.For example the short row 5 2 20 10kh0h000000000 means the following: i) 5 is the column index for A2,3, corresponding to the 5 th orbit of 3-spaces under G. (ii) Augmenting the pair 2 20 with 1, yields the basis of 3 vectors {1, 2, 20} in F 8  2 from which a Fano plane is constructed, and from which the complete 5 th G-orbit can be generated.(iii) By reversing the encoding process discussed earlier, the code "10kh0h000000000" yields the 5 th column of A2,3, as the transpose of 1000110101000101000000000000000000000000000 .
Remark 3.1.In passing, we present some properties of A2,3 which may be used to establish still unknown features of designs and large sets related to A2,3.We say that a vector with integer entries has type (i) The row sums of A2,3 are all 63, as expected, (ii) The vector of column sums of A2,3 is of type 7 360 9 21 , (iii) The row vectors of the long orbits of 2-spaces are all of type 0 320 1 60 3 1 , (iv) the row vector for the short orbit of 2-spaces is of type 0 360 3 21 , (v) There are 4 column types for A2,3 as follows:  In particular, properties (v) d.) and (vi) imply that a large set LS [3][2, 3, 2 8 ] whose automorphism group contains a Singer subgroup as a normal subgroup, can not have a group of automorphisms transitive on the 3 2-[2 8 , 3, 21] designs.

Constructing and presenting the designs and large sets
As the number of columns of A2,3 is rather large, a backtrack, depth-first search or similar algorithm would be hopeless in finding solutions to equation (10).Instead, we use lattice basis reduction to seek solutions.This technique is nicely described in [15], pages 277-300.For each of the 9 large sets of 2-[2 8 , 3, 21] designs we proceed using the following non-deterministic procedure, which, in general, is not guaranteed to terminate.Procedure 3.2.
(ii) Remove from A2,3 the 127 columns corresponding to design D1 to obtain a 43 × 254 matrix C2,3, and find a 0-1 solution u2 to equation C2,3u2 = 21J, thus extracting a second design D2 consisting of 127 G-orbits among the orbits corresponding to the columns of C2,3.If this step succeeds, proceed to step (iii), otherwise stop.
(iii) Remove the 127 columns constituting D2 from C2,3.The remaining 127 columns of C2,3 correspond to orbits whose union is a third design D3, and ] large set.If steps (i) and (ii) are successful, so is (iii) and we have a successful termination with output large set L.
Thus, the procedure of finding u1 and u2 becomes a matter of solving systems of integer equations through lattice basis reduction [15].The following procedure describes briefly how the problems are set up so that lattice basis reduction can be used.Procedure 3.3.First we construct a matrix that will constitute a basis for an integral lattice Λ1 by adjoining the identity matrix of order 381 above KM matrix A2,3.To the right of the 424 × 381 matrix just formed we adjoin a 424 × 1 column vector which has zeros in the first 381 positions and −21's in the remaining 43 positions.Let M1 denote the 424 × 382 matrix just formed.
If basis reduction produces a short enough basis M 1 for Λ1 which contains a short vector v1 with 0's and 1's (or 0's and −1's) in the first 381 positions and all 0's below, then the projection u1 of v1 (or −v1) to the first 381 coordinates is likely to be a solution to A2,3u1 = 21J (see [15].)The weight of u1 will be 127, and the union of orbits of 3-spaces corresponding to the 1's in u1 will form a 2-[2 8 , 3, 21] design D1.
If a solution u1 is found, then replacing A2,3 by C2,3 yields a 297 × 255 matrix M2 which spans a lattice Λ2, and by the same process as above, M2 can yield a solution to C2,3u2 = 21J, that is, a design D2 disjoint from D1.
It is now clear that when the 127 columns corresponding to the orbits forming D2 are removed from C2,3, the remaining 127 orbits will form a 2-[2 8 , 3, 21] design D3, and that {D1, D2, D3} will be a large set.However, the above procedure is not guaranteed to find a solution at first try, so if the basis reduction algorithm was unable to find a column in reduced basis M 1 that met the conditions in Procedure 3.2.2, we would repeat the process, twiking the order of the columns of M1, and the same later for M2.The above procedure was repeated a number of times and we successfully constructed 9 distinct large sets {L1, . . ., L9} which we exhibit below.

Reconstruction of the large sets
We briefly describe the display, to enable the reader to reconstruct the large sets and related designs.The first column is the index of the G-orbits on 3-spaces.There are 9 additional columns, each corresponding to one of the large sets.Each column has 127 1's, 127 2's and 127 3's in it, which select the orbits contained in D1, D2 and D3 respectively, for each large set.Since the orbits can be computed from the representative bases in the presentation of A2,3, the reader can readily reconstruct the 9 large sets and the designs involved.
Direct computation shows that indeed the 9 large sets are different from each other and different from the large set L0 constructed in [4].However, a peculiar visual symmetry is observed in the structure of our 9 large sets Checking the list of maximal subgroups of Γ = GL8 (2) shows that N = NΓ(G) is not maximal in Γ.Let Φ8 = ζ ≤ N , ζ : α → α 2 be the Frobenius subgroup normalizing G.We have checked that Φ8 does not fix any of the 9 large sets, and does not move any one of the 9 large sets to any other.
Let L0 be the LS [3][2, 3, 2 8 ] discovered by the authors of [4], and let S = {Li : 0 ≤ i ≤ 9}.We already know that G ≤ Aut(Li) for each i ∈ {0, . . ., 9}.It is conceivable that the automorphism groups of the 10 large sets in S are not all identical, but we think this is very unlikely and we conjecture that in fact Aut(Li) are all identical, and equal to the Singer subgroup G.

Possible future work
The necessary conditions for the existence of a LS [3]- [3, 4, 2 9 ] are satisfied and we are close to settling the question of existence of a LS [3]- [3, 4, 2 9 ].