Alternative method for construction of Steiner Triple Systems of order n; n≡1 or 3(mod 6)and n>12

International Journal of Research and Scientific Innovation (IJRSI) | Volume VI, Issue XII, December 2019 | ISSN 2321–2705

Alternative method for construction of Steiner Triple Systems of order n; n≡1 or 3(mod 6)and n>12

D.M.T.B. Dissanayake and A.A.I. Perera

Department of Mathematics, Faculty of Science, University of Peradeniya, Sri Lanka

Abstract:-Construction of Steiner Triple System is well-known. In this work, an alternative construction is given for the construction of STS(n); n≡1(mod 6) and n>1. Basic blocks have been used for this construction and these blocks have special properties. Starting with these blocks STS(13), STS(19) and STS(25) have been constructed. Furthermore, generalizations of this work for STS(3n) and STS(n²) have been given by introducing Cartesian Products of two sets.

Keywords: Steiner system, Steiner triple system, Basic Blocks

I. INTRODUCTION

Steiner Systems were introduced by the mathematician Steiner in 1853 and are widely used in constructing designs. A pair (X,B) where X is a n-set and B is a family of m-subsets that any l-set lies in exactly one number of B is called a Steiner System(l,m,n). A Steiner System S(2,3,n) is called a Steiner Triple System of order n and is denoted by STS(n). Construction of STS(n)for n≡1 or 3(mod 6)are well known. One such construction method is using complete graphs K_n. This work gives a recursive construction method of STS(n) using basic blocks as an alternative method. Arecursive construction is given for the construction of STS(13),STS(19),STS(25); n≡1(mod 6). Main focus of this research is to construct triples(blocks) of size three so that each pair of elements are in exactly one block. For this construction, basic blocks B₁,B₂,B₃ etc were constructed by taking the set X of n elements as the additive group Z_n ={0, 1, 2,…, n-2, n-1}.

Definition 1

Let G be an additive group of order v and D is a subset of G of cardinality k. If the set of differences d_i-d_j where d_i, d_j∈D; i≠j contains every non-zero element of G exactly λ times, then D is called a (v,k,λ)-difference set.

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