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1. Balachandran, Niranjan. The 3-Design Problem.

Degree: PhD, Mathematics, 2008, The Ohio State University

This dissertation studies the ‘asymptotic existence’ conjecture for 3-designs with the primary goal of constructing new families of 3-designs. More specifically, this dissertation includes the following: Firstly, by considering the action of the group PSL(2,q) on the finite projective line and the orbits of the action of this group to construct simple 3-designs. While the case q congruent to 3 modulo 4 is 3-homogeneous (so that orbits of any ‘base’ block’ would yield designs), the case q congruent to 1 modulo 4 does not work the same way. We overcome some of these issues by considering appropriate unions of orbits to produce new infinite families of 3-designs with PSL(2,q) acting as a group of automorphisms. We also prove that our constructions actually produce an abundance of simple 3-designs for any block size if q is sufficiently large and also construct a large set of Divisible designs as an application of our constructions. We generalize the notion of a Candelabra system to more general structures, called Rooted Forest Set systems and prove a few general results on combinatorial constructions for these general set structures. Then, we specialize to the case of k=6 and extend a theorem of Hanani to produce new infinite families of Steiner 3-designs with block size 6. Finally, we consider Candelabra systems and prove that a related incidence matrix has full row rank over the rationals. This leads to interesting possibilities for ‘lambda large’ theorems for Candelabra systems. While a ‘lambda large’ theorem for Candelabra systems do not directly yield any Steiner 3-design, it allows for constructions of new Steiner 3-designs on large sets using methods such as Block spreading. Advisors/Committee Members: Robertson, Neil (Advisor).

Subjects/Keywords: Mathematics; 3-design; Candelabra system; projective special linear groups

…application of our constructions. 2. We generalize the notion of a Candelabra system to more general… …block size 6. 3. Finally, we consider Candelabra systems and prove that a related incidence… …for Candelabra systems. While a λ-large theorem for Candelabra systems do not directly yield… …arXiv.org :math/0611842v1), 2006. 3. “New infinite families of Candelabra Systems with block… …size 6 and stem size 2 ”, submitted, 2007. 4. “ A λ-large theorem for Candelabra Systems”, in… 

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Balachandran, N. (2008). The 3-Design Problem. (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1211922186

Chicago Manual of Style (16th Edition):

Balachandran, Niranjan. “The 3-Design Problem.” 2008. Doctoral Dissertation, The Ohio State University. Accessed October 28, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1211922186.

MLA Handbook (7th Edition):

Balachandran, Niranjan. “The 3-Design Problem.” 2008. Web. 28 Oct 2020.

Vancouver:

Balachandran N. The 3-Design Problem. [Internet] [Doctoral dissertation]. The Ohio State University; 2008. [cited 2020 Oct 28]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1211922186.

Council of Science Editors:

Balachandran N. The 3-Design Problem. [Doctoral Dissertation]. The Ohio State University; 2008. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1211922186

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