This article is cited in 1 paper

Coding Theory

Steiner triple systems of order $21$ with a transversal subdesign $\mathrm{TD}(3, 6)$

Y. Guan^a, M. Shi^a, D. S. Krotov<sup>b

a</sup> Key Laboratory of Intelligent Computing and Signal Processing of Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei, Anhui Province, P. R. China
^b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

Abstract: A Steiner triple system (STS) contains a transversal subdesign $\mathrm{TD}(3, w)$ if its point set has three pairwise disjoint subsets $A$, $B$, $C$ of size $w$ and $w^2$ blocks of the STS intersect with each of $A$, $B$, $C$ (those $w^2$ blocks form a $\mathrm{TD}(3, w)$). We prove several structural properties of Steiner triple systems of order $3w + 3$ that contain one or more transversal subdesigns $\mathrm{TD}(3, w)$. Using exhaustive search, we find that there are $2\ 004\ 720$ isomorphism classes of STS(21) containing a subdesign $\mathrm{TD}(3, 6)$ (or, equivalently, a $6 \times 6$ Latin square).

Keywords: Steiner triple system, subdesign, transversal design, Latin square.

UDC: 621.391.1 : 519.1

Received: 20.05.2019
Revised: 23.08.2019
Accepted: 29.08.2019

DOI: 10.31857/S0555292320010039

 English version: Problems of Information Transmission, 2020, 56:1, 23–32

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