Abstract
In this paper, we use the intransitive starter-adder method and the standard starter-adder method to construct some new frames and doubly resolvable group divisible designs. Some infinite classes of frames and doubly resolvable group divisible designs are obtained by recursive constructions. On this basis, we almost establish the existence of frames and doubly resolvable group divisible designs with block size three and index two.
Similar content being viewed by others
Data availability
The authors declare that all data is available.
References
Abel, R.J.R.: Existence of five MOLS of orders 18 and 60. J. Combin. Des. 23, 135–139 (2015)
Abel, R.J.R., Bailey, R.F., Burgess, A.C., Danziger, P., Mendelsohn, E.: On generalized Howell designs with block size three. Des. Codes Cryptogr. 81, 365–391 (2016)
Abel, R.J.R., Bennett, F.E., Greig, M.: PBD-closure. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC handbook of combinatorial designs, 2nd edn., pp. 247–255. CRC Press, Boca Raton (2007)
Abel, R.J.R., Chan, N., Colbourn, C.J., Lamken, E.R., Wang, C., Wang, J.: Doubly resolvable nearly Kirkman triple systems. J. Combin. Des. 21, 342–358 (2013)
Abel, R.J.R., Lamken, E.R., Wang, J.: A few more Kirkman squares and doubly near resolvable BIBDs with block size \(3\). Discrete Math. 308, 1102–1123 (2008)
Chee, Y.M., Kiah, H.M., Zhang, H., Zhang, X.: Constructions of optimal and near-optimal multiply constant- weight codes. IEEE Trans. Inf. Theory 63, 3621–3629 (2017)
Colbourn, C.J., Dinitz, J.H. (eds.): The CRC handbook of combinatorial designs, 2nd edn. CRC Press, Boca Raton (2007)
Colbourn, C.J., Lamken, E.R., Ling, A.C.H., Mills, W.H.: The existence of Kirkman squares-doubly resolvable \((v, 3, 1)\)-BIBDs. Des. Codes Cryptogr. 26, 169–196 (2002)
Colbourn, C.J., Manson, K.E., Wallis, W.D.: Frames for twofold triple systems. Ars Combin. 17, 65–74 (1984)
Colbourn, C.J., Vanstone, S.A.: Doubly resolvable twofold triple systems. Congr. Numer. 34, 219–223 (1982)
Deza, M., Vanstone, S.A.: Bounds for permutation arrays. J. Stat. Plan. Inference 2, 197–209 (1978)
Du, J., Abel, R.J.R., Wang, J.: Some new resolvable GDDs with \(k=4\) and doubly resolvable GDDs with \(k=3\). Discrete Math. 338, 2015–2118 (2015)
Etzion, T.: Optimal doubly constant weight codes. J. Combin. Des. 16, 137–151 (2008)
Ge, G., Miao, Y.: PBDs, frames, and resolvability. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC handbook of combinatorial designs, 2nd edn., pp. 261–265. CRC Press, Boca Raton (2007)
Lamken, E.R.: 3-complementary frames and doubly near resolvable \((v,3,2)\)-BIBDs. Discrete Math. 88, 59–78 (1991)
Lamken, E.R.: The existence of 3 orthogonal partitioned incomplete Latin squares of type \(t^n\). Discrete Math. 89, 231–251 (1991)
Lamken, E.R.: The existence of doubly near resolvable \((v,3,2)\)-BIBDs. J. Combin. Des. 2, 427–440 (1994)
Lamken, E.R.: The existence of doubly resolvable \((v,3,2)\)-BIBDs. J. Combin. Theory Ser. A 72, 50–76 (1995)
Lamken, E.R., Vanstone, S.A.: Existence results for doubly near resolvable \((v, 3, 2)\)- BIBDs. Discrete Math. 120, 135–148 (1993)
Mathon, R., Vanstone, S.A.: On the existence of doubly resolvable Kirkman systems and equidistant permutation arrays. Discrete Math. 30, 157–172 (1980)
Pan, L., Abel, R.J.R., Wang, J.: Constructions of optimal multiply constant-weight codes MCWC \((3\), \(n_1\); \(1, n_2\); \(1, n_3;8)s\). Adv. Math. Commun. (2022). https://doi.org/10.3934/amc.2022025. (Online)
Shi, J., Wang, J.: On existence of two classes of generalized Howell designs with block size three and index two. Graphs Combin. 36, 1525–1543 (2020)
Smith, P.: A doubly divisible nearly Kirkman system. Discrete Math. 18, 93–96 (1977)
Todorov, D.T.: Four mutually orthogonal Latin squares of order 14. J. Combin. Des. 20, 363–367 (2012)
Vanstone, S.A.: Doubly resolvable designs. Discrete Math. 29, 77–86 (1980)
Wilson, R.M.: Constructions and uses of pairwise balanced designs. Math. Centre Tracts 55, 18–41 (1974)
Acknowledgements
The authors are grateful to the referees for their careful reading of the original version of this paper, their detailed comments and the suggestions that much improved the quality of this paper.
Funding
Research supported by the National Natural Science Foundation of China under Grant No. 11671402.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Dong, X., Wang, J. Frames and Doubly Resolvable Group Divisible Designs with Block Size Three and Index Two. Graphs and Combinatorics 39, 109 (2023). https://doi.org/10.1007/s00373-023-02681-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00373-023-02681-5