Correction: On flag-transitive symmetric (v, k, 4) designs

Since the advent of design theory that began with constructive results of R.C. Bose and M. Hall, symmetric designs have occupied a very special status. This is due to several reasons, two most important of which are the structural symmetry and the difficulty in constructions of these designs (compared to other classes of designs). The initial part of this exposition will concentrate on new results on symmetric designs. Quasi-symmetric designs are closely related to symmetric designs. These are designs that have (at the most) two block intersection numbers. One can associate a block graph with a quasi-symmetric design and in many cases of interest, this also turns out to be a graph with special properties, and is called a strongly regular graph. One trivial way of constructing quasi-symmetric designs is to take multiple copies of a symmetric design. Since a symmetric design has exactly one block intersection number, the resulting design will have two block intersection numbers. Such quasi-symmetric designs are called improper. Only one example of a proper quasi-multiple quasi-symmetric design seems to be known so far. The question of the existence of a quasi-multiple of a symmetric design is of some importance particularly when the corresponding symmetric design is known not to exist. In that case, obtaining a quasi-multiple with least multiplicity has attracted attention of combinatorialists in the last thirty years.

Since the advent of design theory that began with constructive results of R.C. Bose and M. Hall, symmetric designs have occupied a very special status. This is due to several reasons, two most important of which are the structural symmetry and the difficulty in constructions of these designs (compared to other classes of designs). The initial part of this exposition will concentrate on new results on symmetric designs. Quasi-symmetric designs are closely related to symmetric designs. These are designs that have (at the most) two block intersection numbers. One can associate a block graph with a quasi-symmetric design and in many cases of interest, this also turns out to be a graph with special properties, and is called a strongly regular graph. One trivial way of constructing quasi-symmetric designs is to take multiple copies of a symmetric design. Since a symmetric design has exactly one block intersection number, the resulting design will have two block intersection numbers. Such quasi-symmetric designs are called improper. Only one example of a proper quasi-multiple quasi-symmetric design seems to be known so far. The question of the existence of a quasi-multiple of a symmetric design is of some importance particularly when the corresponding symmetric design is known not to exist. In that case, obtaining a quasi-multiple with least multiplicity has attracted attention of combinatorialists in the last thirty years.

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This paper is a contribution to the investigation of flag-transitive primitive symmetric (v, k, λ) designs with λ at most 10. We prove that if 𝒟 = (𝒫, ℬ) is a nontrivial symmetric (v, k, λ) design, G ≤ Aut (𝒟) is flag-transitive primitive with Soc (G) = An for n ≥ 5, then 𝒟 is the unique symmetric (35,18,9) design which is the complement of the symmetric (35,17,8) design.

An arrangement of $v$ varieties or treatments in $b$ blocks of size $k, (k < v),$ is known as a balanced incomplete block design if every variety occurs in $r$ blocks and any two varieties occur together in $\lambda$ blocks. These parameters obviously satisfy the equations \begin{equation*}\tag{1} bk = vr\end{equation*}\begin{equation*}\tag{2} \lambda(v - 1) = r(k - 1)\end{equation*} Fisher [1] has also proved that the inequality \begin{equation*}\tag{3} b \geq v, \quad r \geq k\end{equation*} must hold. If $v, b, r, k$ and $\lambda$ are positive integers satisfying (1), (2) and (3), then a balanced incomplete block design with these parameters possibly exists, but the actual existence of a combinatorial solution is not ensured. These conditions are thus necessary but not sufficient for the existence of a design. Fisher and Yates in their tables [2] have listed all designs with $r \leq 10$ and given combinatorial solutions, where known. A balanced incomplete block design in which $b = v$, and hence $r = k$ is called a symmetrical balanced incomplete block design. The impossibility of the symmetrical designs with parameters $v = b = 22, r = k = 7, \lambda = 2$ and $v = b = 29, r = k = 8, \lambda = 2$ was first demonstrated by Hussain [3], [4] essentially by the method of enumeration. The object of the present note is to give an alternative simple proof of the impossibility of these designs and to show that the only unknown remaining symmetrical design in Fisher and Yates' tables, viz. $v = b = 46, r = k = 10, \lambda = 2,$ is definitely impossible. Symmetrical designs with $\lambda \leq 5, r, k \leq 20$, which are impossible combinatorially, are also listed.

The rapid growth of small-size rotorcraft such as Unmanned Aerial Vehicles (UAV's) creates new missions with a new range of issues. Rotor noise is an inevitable consequence of rotary wing flight and can lead to the annoyance or dissatisfaction of customers. This paper presents the experimental work to explore possible acoustic and aerodynamic performance benefits from a proposed anti-phase rotor technology developed previously by NASA Ames and team. The anti-phase alternating pattern from blade to blade aims to prevent harmonic reinforcement of the blade vortex structure that could theoretically lead to an acoustic reduction. A modified NACA-4412 rotor with a NACA-E63 root was used as the baseline rotor for acoustics and aerodynamic performance comparisons. Six 8inch rotors (two sets per design) were manufactured using 3D printing technology. Testing was conducted in the Open Jet Flow-through Anechoic Chamber on the UAV Rotor Test System at Penn State. A semi-circular array that has a radius of 104 cm and held 15 microphones was used to measure the far-field rotor noise. Three flight conditions, hovering and advancing side edgewise flight 9.7 m/s and retreating side edgewise flight at 9.7 m/s, were tested. A total of nine cases for Matching RPM (MR) and nine cases for Matching Thrust (MT) cases were conducted. Possible uncertainties in the study were identified. Recirculation effects of testing in a closed anechoic chamber was acknowledged. A single rotor hover test at Penn State determined that peaks of Sound Pressure Level (SPL) within 2,000-4,000 Hz showed similar values within 10% difference when the test was in the chamber and outside free from recirculation. This range was taken as the range of interest of this study. The repeatability of data between the three runs in each case, showed variations below 10 % in acoustic performance metrics and below 5 % in aerodynamic performance metrics deeming each case repeatable at the point of testing. Physical differences in the advancing and retreating side rotors of the same design caused by uncertainties in manufacturing were identified to have caused discrepancies in the OASPL readings at the compared microphones of up to 2.3 dB. These discrepancy values can be taken as the possible acoustic error value in this study. In hover the modified rotors showed decreased aerodynamic performance and no significant increase in acoustic performance compared to the baseline rotor. In MR cases, the asymmetric and symmetric design had 10 % and 11 % more thrust but required 12.8 % and 8.4 % more torque and had negative values for percentage 1 / Power Loading (1 / PL) respectively. Overall Sound Pressure Level (OASPL) acoustic delta values were up to +1.3 dB louder. For the advancing side of the edgewise flight cases, the asymmetric design had a -5.1% decrease in torque and 8.4 % 1 / PL value, making it a better design for aerodynamic performance as compared to the symmetric design and the baseline rotor. There were also no significant acoustic performance benefits from either modified rotor. The retreating side showed the most significant aerodynamic performance benefits for both modified rotors. In the MR cases, the asymmetric design had a -22.1 % reduction in torque and a percentage 1 / PL value of 27.1 %. At 3,000 Hz, both the symmetric and asymmetric designs demonstrated significant acoustic advantages over the baseline rotor in the MR case. The symmetric design was 4 to 5 dB quieter and the asymmetric design was 3 to 4 dB quieter. This initial experimental exploration of the anti-phase blade concepts showed promising aerodynamic performance and SPL Spectrum at 3,000 Hz acoustic benefits.

From a symmetric balanced incomplete block design we may construct a derived design by deleting a block and its varieties. But a design with the parameters of a derived design may not be embeddable in a symmetric design. Bhattacharya (1) has such an example with λ = 3 . When λ = 1, the derived design is a finite Euclidean plane and this can always be embedded in a corresponding symmetric design which will be a finite projective plane.

Many non-existence theorems are known for symmetric group divisible partial designs. In the case that these partial designs are auto-dual withλ1=0, an ideal incidence structure can be defined whose elements are the equivalence-classes of non-collinear points and parallel blocks. Except for some trivial cases this incidence structure turns out to be a symmetric design, and by studying its existence we can prove much more powerful non-existence theorems.

Let D be a 2-(v;k;4) symmetric design and G be a ∞ag-transitive point-primitive automorphism group of D with X E GAut(X) where X » PSL2(q). Then D is a 2-(15;8;4) symmetric design with X = PSL2(9) and Xx = PGL2(3) where x is a point of D. x1 Introduction A 2-(v;k;‚) design is an incidence structure D = (P;B) where P is a set of v points and B is a set of b blocks with an incidence relation such that every block is incident with exact k points, and every 2-element subset of P is incident with exact ‚ blocks. And D is called a symmetric design if v = b, and is nontrivial if ‚ < k < v i 1. An automorphism of a design D is a permutation of the points which also permutes the blocks. The set of automorphisms of a design with the composition of maps is a group. If G is a primitive permutation group on the point set P then G is called point-primitive, otherwise point-imprimitive. A ∞ag in a design is an incident point-block pair, G is called ∞ag-transitive if G is transitive on the set of ∞ags. There are many works on 2-(v;k;‚) symmetric designs with ‚ small, especially under the condition that the automorphism group of a symmetric design is ∞ag-transitive. For exam- ple, W. M. Kantor in (7) classifled ∞ag-transitive 2-(v;k;1) symmetric designs, which are called projective planes. In (11-14), Regueiro reduced the classiflcation of ∞ag-transitive 2-(v;k;2) sym- metric designs, i.e. biplanes, to the situation where the automorphism group is a 1-dimensional a-ne group. A 2-(v;k;3) symmetric design is called a triplane. In (18-21), Zhou and Dong proved that if D is a nontrivial triplane with a ∞ag-transitive automorphism group G, then D has parameters (11;6;3), (15;7;3), (45;12;3) or G is an a-ne group. Recently, they also clas- sifled the case that the automorphism group is of a-ne type. In (16), Praeger and Zhou have studied ∞ag-transitive, point-imprimitive 2-(v;k;‚) symmetric designs, especially for the cases that ‚ is at most 10. In 2009, Law, Praeger and Reichard classifled ∞ag-transitive (90;20;4)

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CHAPTER 20 - Construction of Block Designs

Metric dimension and metric independence number of incidence graphs of symmetric designs

Symmetric designs with λ ⩽ 10 admitting flag-transitive and point-primitive almost simple automorphism groups