Large sets of Kirkman triple systems of prime power sizes
A Steiner triple system of order \(v\), STS\((v)\), is a pair \(({\mathcal X},{\mathcal B})\), where \(\mathcal X\) is a \(v\)-set and \(\mathcal B\) is a collection of triples (blocks) such that every pair of distinct elements of \(\mathcal X\) is contained in a unique triple of \(\mathcal B\). A STS\((v)\), \(({\mathcal X},{\mathcal B}\)), is said to be resolvable if the triples in \(\mathcal B\) can be partitioned into classes, \(P_1,P_2, \dots, P_{(v-1)/2}\) such that each element of \(\mathcal X\) is contained in precisely one triple of each class. A resolvable STS\((v)\) is called a Kirkman triple system of order \(v\), KTS\((v)\). It is well known that a KTS\((v)\) exists if and only if \(v\equiv 3 \pmod{6}\) [\textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}, Proc. Sympos. Pure Math. 19, 187--203 (1971; Zbl 0248.05009)]. Two STS\((v)\)s on the same set \(\mathcal X\) are said to be disjoint if they have no triples in common. A partition of all triples of a \(v\)-set into pairwise disjoint STS\((v)\)s is called a large set of STS\((v)\)s, denoted by LSTS\((v)\). An LSTS\((v)\) consists of \(v-2\) pairwise disjoint STS\((v)\)s. A large set of Kirkman triple systems is denoted by LKTS\((v)\), where each STS\((v)\) is a KTS\((v)\). In [Des. Codes Cryptography 83, No. 1, 23--32 (2017; Zbl 1362.05023)], the authors constructed LKTS\((q +2)\) for any admissible prime power \(q < 400\) with 12 possible exceptions. In this paper, they extend this work and provide direct constructions for 10 of these remaining cases. Their main construction, Theorem 2.2, describes the properties required for a difference family of type \((q-1, \{2,3\},1)\) to exist that will produce an LKTS\((q+2)\). They use computer searches to construct 4 designs using Theorem 2.2. They extend work in two previous papers to provide direct constructions (again based on difference families) to take care of 6 more cases. They note that their computer searches and algorithms can also be applied to construct 5 LKTSs for larger values of \(v\).
scientific article; zbMATH DE number 6798855
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