Enumeration of symmetric ( 45 , 12 , 3 ) designs with nontrivial automorphisms ∗

We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. Further, we discuss trigeodetic graphs obtained from the symmetric (45, 12, 3) designs. We prove that k-geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs obtained from symmetric (45, 12, 3) designs. 2010 MSC: 05B05, 20D45, 94B05, 05C38


Introduction
The terminology and notation in this paper for designs and codes are as in [2,3,6].
One of the main problems in design theory is that of classifying structures with given parameters.Classification of designs has been considered in detail in the monograph [17].Complete classification of designs with certain parameters has been done just for some designs with relatively small number of points, and in the case of symmetric designs complete classification is done just for a few parameter triples (see [22]).The classification of projective planes of order 9 has been solved in 1991 (see [20]), and Kaski and Östergård classified all biplanes with k=11 in 2008 (see [18]).Hence, the parameter triple (45,12,3) is the next for symmetric designs of order 9 to be classified.Since the complete classification of symmetric (45,12,3) designs seems to be out of reach with the current techniques and computers, only partial classification of such designs, with certain constrains, is possible.In this paper we manage to classify all symmetric (45,12,3) designs with nontrivial automorphisms.
In this paper we give the classification of all symmetric (45,12,3) designs having a nontrivial automorphism group.We show that there exist exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms, which means that there are at least 5421 symmetric (45,12,3) designs.Furthermore, we discuss trigeodetic graphs obtained from the symmetric (45,12,3) designs and prove that mutually non-isomorphic designs produce mutualy non-isomorphic k-geodetic graphs.
The paper is organized as follows: after the brief introduction, in Section 2 we give basic information concerning the construction method, in Section 3 we describe the construction of symmetric (45,12,3) designs with nontrivial automorphisms and give a list of the designs and their full automorphism groups, Section 4 gives information about the codes of the constructed designs, and in Section 5 we discuss trigeodetic graphs obtained from the symmetric (45, 12, 3) designs.
For the construction of designs we have used our own computer programs.For isomorphism testing, and to obtain and analyze the full automorphism groups of the designs we have used [14] and [30].The codes have been analyzed using Magma [4].

Outline of the construction
An incidence structure D = (P, B, I), with point set P, block set B and incidence I is a t-(v, k, λ) design, if |P| = v, every block B ∈ B is incident with precisely k points, and every t distinct points are together incident with precisely λ blocks.A design is called symmetric if it has the same number of points and blocks.An automorphism of a design D is a permutation on P which sends blocks to blocks.The set of all automorphisms of D forms its full automorphism group denoted by Aut(D).
Let D = (P, B, I) be a symmetric (v, k, λ) design and G ≤ Aut(D).The group action of G produces the same number of point and block orbits (see [21,Theorem 3.3]).We denote that number by t, the point orbits by P 1 , . . ., P t , the block orbits by B 1 , . . ., B t , and put |P r | = ω r and |B i | = Ω i .An automorphism group G is said to be semi-standard if, after possibly renumbering orbits, we have ω i = Ω i , for i = 1, . . ., t.We denote by γ ir the number of points of P r which are incident with a representative of the block orbit B i .For these numbers the following equalities hold (see [5,9,16]): Definition 2.1.A (t × t)-matrix (γ ir ) with entries satisfying conditions (1) and (2) is called an orbit matrix for the parameters (v, k, λ) and orbit lengths distributions (ω 1 , . . ., ω t ), (Ω 1 , . . ., Ω t ).
The construction of designs admitting an action of a presumed automorphism group, using orbit matrices, consists of the following two basic steps (see [5,9,16]): 1. Construction of orbit matrices for the given automorphism group, 2. Construction of block designs for the orbit matrices obtained in this way.This step is often called an indexing of orbit matrices.
In order to construct the orbit matrices for an action of a presumed automorphism group we have to determine all possibilities for the orbit lengths distributions.The following facts, that one can use in that purpose, can be found in [21].
Theorem 2.2.An automorphism ρ of a symmetric design fixes an equal number of points and blocks.Moreover, ρ has the same cyclic structure, whether considered as a permutation on points or on blocks.
Theorem 2.3.Suppose that a nonidentity automorphism ρ of a nontrivial symmetric (v, k, λ) design fixes f points.Then where n = k − λ is the order of the design.Moreover, if equality holds in either inequality, ρ must be an involution and every non-fixed block contains exactly λ fixed points.
Theorem 2.4.Suppose that D is a nontrivial symmetric (v, k, λ) design, with an involution ρ fixing f points and blocks.If f = 0, then Suppose that D is a symmetric (v, k, λ) design with an automorphism ρ of prime order p fixing f points.Then f ≡ v (mod p), and ρ acts semi-standardly on D. In that case, since the action of G = ρ is semi-standard, it is sufficient to determine point orbit lengths distribution (ω 1 , . . ., ω t ).After determining the orbit lengths distributions we proceed with the construction of orbit matrices and corresponding designs, as described in [9].

Symmetric (45,12,3) designs admitting Z 2 as an automorphism group
Let ρ be an involutory automorphism of a symmetric (45, 12, 3) design fixing f points.Then 5 ≤ f ≤ 15 and f ≡ 1 (mod 2), hence f ∈ {5, 7, 9, 11, 13, 15}.Up to isomorphism there are 682 orbit structures, that produce 2987 mutually non-isomorphic designs.Information about the number of the orbit structures and the designs are given in Table 1.It was determined in [8] that there are exactly 591 orbit matrices for the group Z 3 acting on symmetric (45,12,3) designs.From these orbit matrices we have obtained up to isomorphism exactly 2108 symmetric (45,12,3) designs that admit an automorphism of order three.Information about the number of the orbit matrices and the constructed designs are presented in Table 2. Comparing the designs described in subsections 3.1 and 3.2 we conclude that up to isomorphism there are exactly 4280 symmetric (45,12,3) designs that admit an automorphism of order 2 or 3.It is known from [5,19,25] that there are exactly 13 symmetric (45,12,3) designs with an automorphism of order 5, and only four of them have the full automorphism group whose order is not divisible by 2 or 3. Further, there is exactly one symmetric (45,12,3) design with an automorphism of order 11, and the full automorphism group of that designs is Z 11 .That shows that there exist exactly 4285 symmetric (45,12,3) designs with a nontrivial automorphism group.Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs.Information about these 4285 designs and their full automorphism groups are given in Table 3.Some od the automorphism groups have the same description of the structure, but they are not isomorphic.In that case, nonisomorphic groups with the same structure are listed in separate rows of Table 3 (e.g. two groups of order 324 having the structure (E 27 : Z 3 ) : E 4 ).Since Mathon and Spence have constructed 1136 symmetric (45,12,3) designs with a trivial automorphism group (see [25]), we conclude that up to isomorphism there are at least 5421 symmetric (45,12,3) designs.

Ternary codes from symmetric (45,12,3) designs
The code C F (D) of a design D = (P, B, I) over the finite field F is the space spanned by the incidence vectors of the blocks over F .If Q is any subset of the point set P, then we will denote the incidence vector of Q by and is a subspace of F P , the full vector space of functions from P to F .The following theorem, that can be found in [2], shows that the code C F (D) over a field F of characteristic p is not interesting if p does not divide the order of D. In Theorem 4.2 rank p (D) denotes the dimension of C F (D), and j denotes the all-one vector.Since the order of a symmetric (45,12,3) design is 9, we consider only the ternary codes of the constructed designs, i.e. codes over the field of order 3.The ternary codes of the 4285 symmetric (45,12,3) designs with nontrivial automorphisms are divided in 1005 equivalence classes.In Table 4 we give information about code parameters and orders of automorphism groups of representatives of equivalence classes, where the definitions of automorphisms and equivalence of codes are the same as in Magma [4].The following theorem states that all the codes obtained are self-orthogonal.In Table 5 we give information about the dual codes of the codes presented in Table 4.According to [15] and [26], the [45,28,8] code has the greatest minimum distance among the known ternary [45,28] codes.Further, the best known ternary [45,30] code has minimum distance 7, hence the [45,30,6] code has minimum distance one less than the best known code.
A linear code whose dual code supports the blocks of a t-design admits one of the simplest decoding algorithms, majority logic decoding (see [28]).If a codeword x = (x 1 , . . ., x n ) ∈ C is sent over a communication channel, and a vector y = (y 1 , . . ., y n ) is received, for each symbol y i a set of values y (1) i , . . ., y (ri) i of r i linear functions defined by the blocks of the design are computed, and y i is decoded as the most frequent among the values y errors by majority logic decoding, where r = λ n−1 w−1 .
Consequently, the codes listed in Table 5 can correct up to two errors by majority logic decoding.or 5 with vertex degrees r and k, in which there are at most µ paths of minimum length between any pair of vertices, where . ., B b being blocks of the design (see [29]).
K * v (r, k, λ) has v(r + 1) vertices and vr(k+1) 2 edges.If D is a symmetric design then K * v (r, k, λ) is k−regular graph in which there are at most λ paths of minimum length between each pair of vertices.Graphs in which every pair of nonadjacent vertices has a unique path of minimum length between them are called geodetic graph, bigeodetic graphs are graphs in which each pair of nonadjacent has at most two paths of minimum length between them and graphs in which each pair of nonadjacent vertices has at most k paths of minimum length between them are called k-geodetic graphs (see [13], [29]).
for the design D. The adjacency matrix of a graph K * v (r, k, λ) is given as follows , and m r,s = 0 otherwise, 0 k is the k × k zero-matrix, J k is the k × k all-one matrix, and I k is the k × k identity matrix.Rows of M i , 1 ≤ i ≤ b, are labeled with x i1i , ..., x i k i , and columns of M i , 1 ≤ i ≤ b, are labeled with x 10 , ..., x v0 .The number of columns labeled with x s0 , 1 ≤ s ≤ v in which matrices M i and M j both have an entry 1 is equal to |B i ∩ B j |, since in the column x s0 there is an entry 1 in both matrices if and only if P s ∈ B i ∩ B j .Moreover, the matrix M i is determined by the i th row of the incidence matrix IM = [d i,s ] of the design D. Vice versa, the i th row of the incidence matrix IM = [d i,s ] is determined by the matrix M i , putting d i,s = 1 if there exists a row of M i having 1 on the position x s0 .
Conversely, each isomorphism from the graph K * v (r, k, λ) 1 onto K * v (r, k, λ) 2 induces unique isomorphism from the design D 1 onto D 2 .To prove this statement it is crusial to show that an isomorphism from If the designs D 1 and D 2 are not symmetric, then r = k and since the vertices x 1 10 , . . ., x 1 v0 and x 2 10 , . . ., x 2 v0 have degree r and the other vertices of K * v (r, k, λ) 1 and K * v (r, k, λ) 2 have degree k, it is clear that an isomorphism from K * v (r, k, λ) 1 onto K * v (r, k, λ) 2 maps the set {x 1 10 , . . ., x 1 v0 } onto {x 2 10 , . . ., x 2 v0 }.
If D 1 and D 2 are symmetric designs then r = k.A vertex x 1 i0 and a vertex adjacent to x 1 i0 have no common neighbour, while a vertex that do not belong to {x 1 10 , . . ., x 1 v0 } has k − 2 commmon neighbours with any of its neighbour.Similarly, a vertex x 2 i0 and a vertex adjacent to him have no common neighbour, while a vertex that do not belong to {x 2 10 , . . ., x 2 v0 } has k − 2 commmon neighbours with any of its neighbour.Hence, we conclude that {x 1 10 , . . ., x 1 v0 } is mapped onto {x 2 10 , . . ., x 2 v0 }.So, an isomorphism from K * v (r, k, λ) 1 onto K * v (r, k, λ) 2 maps (K k ) , and M 1 i onto M 2 j , and it induces unique isomorphism from the design D 1 onto D 2 .

Theorem 4 . 1 .
Let D = (P, B, I) be a nontrivial 2-(v, k, λ) design of order n.Let p be a prime and let F be a field of characteristic p, where p does not divide n.Thenrank p (D) ≥ (v − 1)with equality if and only if p divides k; in the case of equality we have that C F (D) = j ⊥ and otherwise C F (D) = F P .

Theorem 4 . 2 .Table 4 .
Let D be a symmetric(45,12,3) design and C(D) be the ternary code of the design D. Then the code C(D) is self-orthogonal, and j ∈ C(D) ⊥ .Proof.The code C(D) is spanned by the rows of the row-point incidence matrix of D. Since each row of D has 12 points, and any two blocks intersect in 3 points, the code C(D) is self-orthogonal.It is obvious that j ∈ C(D) ⊥ , because each row of the design D consist of 12 points.Ternary codes of the symmetric (45,12,3) designs with nontrivial automorphisms Parameters (|Aut(C)|, no. of inequivalent codes)

.
The following result have been obtained by Rudolph[28].Theorem 4.3.If C is a linear [n, k] code such that C ⊥ contains a set S of vectors of weight w whose supports are the blocks of a 2-(n, w, λ) design, the code C can correct up to e = r + λ − 1 2λ

Theorem 5 . 2 .
Let D be a 2-(v, k, λ) design.Then the full automorphism group of D is isomorphic to the full automorphism group of the corresponding k-geodetic graph K * v (r, k, λ).