### On The Non-Existence of a Symmetric Nearly Triply Regular 2 - (78, 22, 6) Design with Μ = 2 And U = 0

Blessilda P. Raposa

**Discipline: **Mathematics

#### Abstract:

A t-(v, k, A) design or t-design consists of a set P of v points and a set B of k-subsets of P called blocks such that any t-subset of Plies in exactly </em>A <em>blocks. By a design, we will mean a 2-design. A design is symmetric if! Pl= /Bl. We say that a symmetric design is nearly triply regular (NTR) if there exist non-negative integers μ and v such that I ex n /3 n </em>r <em>I </em>= <em>for </em>e <em>where </em>A.~μ> <em>v </em>2 <em>o. In this paper, we prove by construction that there does not exist a symmetric NTR2-(78,22, 6) design with μ </em>= <em>2 and v </em>= <em>0. </em>The concept of nearly triple regularity was first introduced by M. Herzog and K. B. Reid {Herzog and Reid 1 97 6). It was originally applied to a certain family of graphs called Hadamard tournaments. In 1989, N. Ito (Ito and Raposa 1992) generalized this concept as Hadamard designs and eventually as the general symmetric design. A year later, C. E. Praeger further extended the concept to non-symmetric designs.

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