Abstract
Non-uniform group divisible designs have been studied by numerous researchers in the past two decades due to their vital applications in the constructions for other types of designs. Much progress has been made for the existence of \(\{4\}\)-GDDs of type \(g^um^1\), especially when \(gu\) is even. The corresponding problem for block size three had been solved by Colbourn et al. (J Comb Theory Ser A 59:73–89, 1992). In this paper, we consider the entire existence problem for such \(\{4\}\)-GDDs. We show that, for each given g, up to a small number of undetermined cases of \(u\), the necessary conditions on \((u,m)\) for the existence of a \(\{4\}\)-GDD of type \(g^um^1\) are also sufficient.
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Acknowledgments
Research supported by the National Natural Science Foundation of China under Grant No. 61171198 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ13A010001. The authors express their gratitude to the two anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of the technical presentation of this paper.
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Communicated by C. J. Colbourn.
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Wei, H., Ge, G. Group divisible designs with block size four and group type \(g^um^1\) . Des. Codes Cryptogr. 74, 243–282 (2015). https://doi.org/10.1007/s10623-013-9854-z
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DOI: https://doi.org/10.1007/s10623-013-9854-z