Abstract
A Skolem-type sequence is a sequence (s 1, . . . , s t ) 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 s j = s j+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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Communicated by C. J. Colbourn.
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Silvesan, D., Shalaby, N. Cyclic block designs with block size 3 from Skolem-type sequences. Des. Codes Cryptogr. 63, 345–355 (2012). https://doi.org/10.1007/s10623-011-9559-0
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DOI: https://doi.org/10.1007/s10623-011-9559-0