Cyclic block designs with block size 3 from Skolem-type sequences

A Skolem-type sequence is a sequence (s1, . . . , st) of positive integers $${i\in D}$$ such that for each $${i\in D}$$ there is exactly one $${j\in \{1, \ldots , t - i\}}$$ such that sj = sj+i = i. Positions in the sequence not occupied by integers $${i\in D}$$ contain null elements. In 1939, Peltesohn solved the existence problem for cyclic Steiner triple systems for v ≡ 1, 3(mod 6), v ≠ 9. Using the same technique in 1981, Colbourn and Colbourn extended the solution to all admissible λ > 1. It is known that Skolem-type sequences may be used to construct cyclic Steiner triple systems as well as cyclic triple systems with λ = 2. The main result of this paper is an extension of former results to cyclic triple systems with λ > 2. In addition we introduce a new kind of Skolem-type sequence.

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