engleski

Hadamard 2-(47, 23, 11) Designs having $Frob_{;55};$ as an Automorphism Group

A symmetric $((v, k, \lambda)$ design is a finite incidence structure $({;\mathcal{;P};, \mathcal{;B};};, I)$, where ${;\mathcal{;P};};$ and ${;\mathcal{;B};};$ are disjoint sets and $I\subseteq {;\mathcal{;P};}; \times {;\mathcal{;B};};$, with the following properties: [1.] $|{;\mathcal{;P};};|=|{;\mathcal{;B};};|=v$, [2.] every element of ${;\mathcal{;B};};$ is incident with exactly $k$ elements of ${;\mathcal{;P};};$, [3.] every pair of elements of ${;\mathcal{;P};};$ is incident with exactly $\lambda$ elements of ${;\mathcal{;B};};$. Elements of the set ${;\mathcal{;P};};$ are called points and elements of the set ${;\mathcal{;B};};$ are called blocks. For two designs ${;\cal D};_1= ({;\mathcal{;P};_1, \mathcal{;B};_1};, I_1)$ and ${;\cal D};_2= ({;\mathcal{;P};_2, \mathcal{;B}; ; _2};, I_2)$ an isomorphism from ${;\cal D};_1$ onto ${;\cal D};_2$ is a bijection which maps points onto points and blocks onto blocks preserving the incidence. An isomorphism from ${;\cal D};$ onto ${;\cal D};$ is an automorphism of ${;\cal D};$. Set of all automorphisms of a design ${;\cal D};$ is called a full automorphism group of ${;\cal D};$ and will be denoted by $Aut\/{;\cal D};$. Let ${;\mathcal{;D};=(\mathcal{;P};, \mathcal{;B};};, I)$ be a symmetric $(v, k, \lambda )$ design and $G\leq Aut{;\mathcal{;D};};$. Let us denote the point orbits by ${;\mathcal{;P};};_{;1};, \ldots , {;\mathcal{;P};};_{;t};$, block orbits by ${;\mathcal{;B};};_{;1};, \ldots , {;\mathcal{;B};};_{;t};$ , and put $|{;\mathcal{;P};};_{;r};|=\omega _{;r};$, $|{;\mathcal{;B};};_{;i};|=\Omega _{;i};$. We shall denote by $\gamma_{;ir};$ the number of points of ${;\mathcal{;P};};_{;r};$ which are incident with the representative of the block orbit ${;\mathcal{;B};};_{;i};$. The following equalities hold: \begin{;eqnarray}; \sum _{;r=1};^{;t};{;\gamma };_{;ir}; & = & k\, , \\ \sum _{;r=1};^{;t};\frac{;{;\Omega };_{;j};};{;{;\omega };_{;r};};{;\gamma };_{;ir};{;\gamma };_{;jr}; & = & \lambda {;\Omega };_{;j};+{;\delta };_{;ij};\cdot (k-\lambda )\, . \end{;eqnarray}; ; DEFINITION The $(t \times t)$-matrix $({;\gamma};_{;ir};)$ with entries satisfying properties (1) and (2) is called the orbit structure for parameters $(v, k, \lambda)$ and orbit distribution $(\omega_{;1};, \ldots , \omega_{;t};)$, $(\Omega_{;1};, \ldots , \Omega_{;t};)$. Construction of symmetric block designs admitting an action of presumed automorphism group consists of two basic steps: (1) construction of orbit structures for given automorphism group, (2) construction of symmetric block designs for obtained orbit structures. Because of the large number of possibilities, it is often necessary to involve a computer in both steps of the construction. We use elements of normalizer of presumed automorphism group to eliminate isomorphic structures in both steps during the construction. Using this method, we have proved the following statement: Up to isomorphisms there are precisely 54 symmetric designs with parameters (47, 23, 11) admitting a faithful action of a Frobenius group of order 55. Six of these designs have $Frob_{;55};$ as a full automorphism group, 39 designs have $Frob_{;55}; \times Z_2$ and 6 designs have $Frob_{;55}; \times S_3$ as a full automorphism group. Full automorphism group of three designs is isomorphic to the group $L_2(11) \times S_3$ of order 3960. Theese designs are not isomorphic to the only one previously known symmetric (47, 23, 11) design, which had been constructed via a cyclic difference set. From those 54 designs we have constructed 179 pairwise nonisomorphic 2-(23, 11, 10) designs as derived and 191 pairwise nonisomorphic 2-(24, 12, 11) designs as residual designs. Among them there are 3 quasi-symmetric 2-(23, 11, 10) designs and 3 quasi-symmetric 2-(23, 12, 11) designs.

block design; Hadamard matrix; automorphism group

nije evidentirano