On quasi-symmetric designs with intersection difference three

Mavron, V. C.; McDonough, T. P.; Shrikhande, Mohan S.

doi:10.1007/s10623-011-9536-7

Cited by 8 publications

(7 citation statements)

References 22 publications

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“…Some works have been done on trianglefree quasi-symmetric 2-designs for the intersection numbers 0 and y in [8]. Later, many works have been developed in [10], [12] and [13]. We present here some of the relevant results.…”

Section: Theorem 4 Let D Be a Quasi-symmetric Design With Standard Pmentioning

confidence: 97%

“…During the last several decades quasi-symmetric 2-designs and their classification play an important role to the study of design theory see for example [6,10,12,13,15,16,18]. Many results have been developed in the theory of binary codes, basically on self-complementary codes, self-dual codes using such designs.…”

Section: Introductionmentioning

confidence: 99%

“…In [5], several necessary conditions are obtained for the existence of a quasisymmetric 2-design with parameter set D (v, b, r, k, λ; x, y) by imposing the divisibility restrictions on y − x. It was also shown in [16] that there are finitely many such designs for y ≥ 2 and fixed block size k. Most of the recent works on quasi-symmetric 2-designs have been concentrated on the difference of the intersection numbers x and y, where y = x + 2 in [14] and [10] i.e. the difference of the intersection numbers is 2 and 3.…”

Section: Introductionmentioning

confidence: 99%

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On some parametric classifications of quasi-symmetric 2-designs

Ghosh¹,

Dey²

2015

Tamkang J. Math.

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Section: Theorem 4 Let D Be a Quasi-symmetric Design With Standard Pmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On some parametric classifications of quasi-symmetric 2-designs

Ghosh¹,

Dey²

2015

Tamkang J. Math.

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“…The classification of triangle-free QS 3-designs is given in [24]. Further investigation of triangle-free QSDs with non-zero intersection numbers are carried out in [21], [25], [26], [27], [29]. Considerable evidence is presented in support of the conjecture that the only triangle-free QSDs with x > 0 are the complements of QSDs with x = 0.…”

Section: Tmentioning

confidence: 99%

Conditions for the Parameters of the Block Graph of Quasi-Symmetric Designs

Pawale

Shrikhande

Nyayate³

2015

Electron. J. Combin.

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A quasi-symmetric design (QSD) is a 2-$(v,k,\lambda)$ design with intersection numbers $x$ and $y$ with $x< y$. The block graph of such a design is formed on its blocks with two distinct blocks being adjacent if they intersect in $y$ points. It is well known that the block graph of a QSD is a strongly regular graph (SRG) with parameters $(b,a,c,d)$ with smallest eigenvalue $ -m =-\frac{k-x}{y-x}$.The classification result of SRGs with smallest eigenvalue $-m$, is used to prove that for a fixed pair $(\lambda\ge 2,m\ge 2)$, there are only finitely many QSDs. This gives partial support towards Marshall Hall Jr.'s conjecture, that for a fixed $\lambda\ge 2$, there exist finitely many symmetric $(v, k, \lambda)$-designs.We classify QSDs with $m=2$ and characterize QSDs whose block graph is the complete multipartite graph with $s$ classes of size $3$. We rule out the possibility of a QSD whose block graph is the Latin square graph $LS_m (n)$ or complement of $LS_m (n)$, for $m=3,4$.SRGs with no triangles have long been studied and are of current research interest. The characterization of QSDs with triangle-free block graph for $x=1$ and $y=x+1$ is obtained and the non-existence of such designs with $x=0$ or $\lambda > 2(x+2)$ or if it is a $3$-design is proven. The computer algebra system Mathematica is used to find parameters of QSDs with triangle-free block graph for $2\le m \le 100$. We also give the parameters of QSDs whose block graph parameters are $(b,a,c,d)$ listed in Brouwer's table of SRGs.

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“…A more recent characterization of the geometric designs PG 2 (4, q) in terms of good blocks 3 -a notion introduced in [21]-is due to Mavron, McDonough and Shrikhande [20]. Their result characterizes the geometric design PG 2 (4, q) among all quasi-symmetric designs with the same parameters and with intersection numbers 1 and q + 1 by the property that all blocks of the design are good.…”

Section: Introductionmentioning

confidence: 99%

Characterizing geometric designs, II

Jungnickel

2011

Journal of Combinatorial Theory, Series A

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We provide a characterization of the classical point-line designs PG 1 (n, q), where n 3, among all non-symmetric 2-(v, k, 1)-designs as those with the maximal number of hyperplanes. As an application of this result, we characterize the classical quasisymmetric designs PG n−2 (n, q), where n 4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q + 1 and all intersection numbers at least q n−4 + · · · + q + 1. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG 1 (n, q); in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.

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On quasi-symmetric designs with intersection difference three

Cited by 8 publications

References 22 publications

On some parametric classifications of quasi-symmetric 2-designs

On some parametric classifications of quasi-symmetric 2-designs

Conditions for the Parameters of the Block Graph of Quasi-Symmetric Designs

Characterizing geometric designs, II

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