An algebraic representation of Steiner triple systems of order 13

Pavone, Marco

doi:10.1016/j.exco.2021.100013

Cited by 2 publications

(1 citation statement)

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“…It is worth mentioning that algebraic representations of the STS(13)s were recently given in [31] and in [12,Proposition 4]. In the former case, an incidence structure isomorphic to a Steiner triple system of order 13 is constructed by defining a set B of twenty-six vectors in the 13-dimensional vector space V over the finite field F 5 , with the property that there exist precisely thirteen 6-subsets of B whose elements sum up to zero in V, whereas in the latter case the STS(13)s are described as sections of A(13)/A (12) in the alternating group A(13).…”

Section: The Two Steiner Triple Systems Of Order 13mentioning

confidence: 99%

A visual representation of the Steiner triple systems of order 13

Pavone

2023

Art Discrete Appl. Math.

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Steiner triple systems (STSs) are a basic topic in combinatorics. In an STS the elements can be collected in threes in such a way that any pair of elements is contained in a unique triple. The two smallest nontrivial STSs, with 7 and 9 elements, arise in the context of finite geometry and nonsingular cubic curves, and have well-known pictorial representations. On the contrary, an STS with 13 elements does not have an intrinsic geometric nature, nor a natural pictorial illustration. In this paper we present a visual representation of the two non-isomorphic Steiner triple systems of order 13 by means of a regular hexagram. The thirteen points of each system are the vertices of the twelve equilateral triangles inscribed in the hexagram. In the case of the non-cyclic system, our representation allows one to visualize in a simple, elegant and highly symmetric way the twenty-six triples, the six automorphisms and their orbits, the eight quadrilaterals, the ten mitres, the thirteen grids, the four 3-colouring patterns, the block-colouring and some distinguished ovals. Our construction is based on a very simple idea (seeing the blocks as much as possible as equilateral triangles), which can be further extended to get new representations of the STSs of order 7 and 9, and of one of the STSs of order 15.

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Section: The Two Steiner Triple Systems Of Order 13mentioning

confidence: 99%