Abstract
A directed triplewhist tournament on p players over Z p is said to have the three-person property if no two games in the tournament have three common players. We briefly denote such a design as a 3PDTWh(p). In this paper, we investigate the existence of a Z-cyclic 3PDTWh(p) for any prime p ≡ 1 (mod 4) and show that such a design exists whenever p ≡ 5, 9, 13 (mod 16) and p ≥ 29. This result is obtained by applying Weil’s theorem. In addition, we also prove that a Z-cyclic 3PDTWh(p) exists whenever p ≡ 1 (mod 16) and p < 10, 000 except possibly for p = 257, 769.
Similar content being viewed by others
References
Abel RJR, Bennett FE, Ge G, Existence of directed triplewhist tournaments with the three person property 3PDTWh(v), preprint
Abel RJR, Finizio NJ, Ge G and Greig M (2006). New Z-cyclic triplewhist frames and triplewhist tourment designs. Discrete Appl Math 154: 1649–1673
Abel RJR and Ge G (2005). Some difference matrix constructions and an almost completion for the existence of triplewhist tournaments TWh(v). European J Combin 26: 1094–1104
Anderson I (1995). A hundred years of whist tournaments. J Combin Math Combin Comput 19: 129–150
Anderson I and Finizio NJ (1997). Triplewhist tournaments that are also Mendelsohn designs. J Combin Des 5: 397–406
Anderson I and Finizio NJ (2000). On the construction of directed triplewhist tournaments. J Combin Math Combin Comput 35: 107–115
Anderson I, Finizio NJ and Leonard P (1999). New product theorems for Z-cyclic whist tournaments. J Combin Theory Ser A 88: 162–166
Baker RD (1975) Factorization of graphs, Doctoral Thesis, Ohio State University
Bennett FE and Ge G (2006). Existence of directedwhist tournaments with the three person property 3PDWh(v). Discrete Appl Math 154: 1939–1946
Bennett FE and Zhu L (1992). Conjugate-orthogonal latin squares and related structures. In: Dinitz, J and Stinson, D (eds) Contemporary design theory: a collection of surveys, pp 41–96. Wiley, New York
Beth T, Jungnickel D and Lenz H (1999). Design theory. Cambridge University Press, Cambridge, UK
Buratti M (2000). Existence of Z-cyclic triplewhist tournaments for a prime number of players. J Combin Theory Ser A 90: 315–325
Buratti M (2002). Cyclic designs with block size 4 and related optimal optical orthogonal codes. Des Codes Cryptogr 26: 111–125
Chang Y and Ji L (2004). Optimal (4up,5,1) optical orthogonal codes. J Combin Des 5: 135–146
Chen K (1995). On the existence of super-simple (v,4,3)-BIBDs. J Combin Math Combin Comput 17: 149–159
Chen K (1996). On the existence of super-simple (v,4,4)-BIBDs. J Statist Plann Inference 51: 339–350
Chen K and Zhu L (1998). Existence of (q,6,1) difference families with q a prime power. Des Codes Cryptogr 15: 167–173
Feng T and Chang Y (2006). Existence of Z-cyclic 3PTWh(p) for any prime p ≡ 1 (mod 4). Des Codes Cryptogr 39: 39–49
Finizio NJ (1993). Whist tournaments–three person property. Discrete Appl Math 45: 125–137
Finizio NJ (1995). Z-cyclic triplewhist tournaments-the noncompatible case, Part 1. J Combin Des 3: 135–146
Finizio NJ (1997). Z-cyclic triplewhist tournaments-the noncompatible case, Part 2. J Combin Des 5: 189–201
Finizio NJ and Lewis JT (1997). A criterion for cyclic whist tournaments with the three person property. Util Math 52: 129–140
Ge G Triplewhist tournaments with the three person property. J Combin Theory Ser A (to appear)
Ge G and Lam CWH (2003). Some new triplewhist tournaments TWh(v). J Combin Theory Ser A 101: 153–159
Ge G and Lam CWH (2004). Super-simple resolvable balanced incomplete block designs with block size 4 and index 3. J Combin Des 12: 1–11
Ge G and Lam CWH (2004). Whist tournaments with the three person property. Discrete Appl Math 138: 265–276
Ge G and Ling ACH (2003). A new construction for Z-cyclic whist tournaments. Discrete Appl Math 131: 643–650
Ge G and Zhu L (2001). Frame constructions for Z-cyclic triplewhist tournaments. Bull Inst Combin Appl 32: 53–62
Gronau H-DOF and Mullin RC (1992). On super-simple 2-(v,4,λ) designs. J Combin Math Combin Comput 11: 113–121
Hartman A (1980) Doubly, orthogonally resolvable quadruple systems, combinatorial mathematics, VII (Proc. Seventh Australian Conf., Univ. Newcastle, Newcastle, 1979), pp 157–164. Lecture Notes in Math. 829, Springer, Berlin
Liaw YS (1996). Construction of Z-cyclic triplewhist tournaments. J Combin Des 4: 219–233
Lidl R and Niederreiter H (1997). Finite fields.. Cambridge University Press, Cambridge, UK
Lu Y and Zhang S (2000). Existence of whist tournaments with the three-person property 3PWh(v). Discrete Appl Math 101: 207–219
Lu Y and Zhu L (1997). On the existence of triplewhist tournaments TWh(v). J Combin Des 5: 249–256
Moore EH (1896). Tactical memoranda I–III. Amer J Math 18: 264–303
Zhang X (1996). On the existence of (v, 4, 1)-RPMD. Ars Combin 42: 3–31
Zhang X (2005). A few more RPMDs with k = 4. Ars Combin 74: 187–200
Zhang X and Ge G (2007). Super-simple resolvable balanced incomplete block designs with block size 4 and index 2. J Combin Des 15: 341–356
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: C.J. Colbourn.
Gennian Ge’s Research was supported by National Natural Science Foundation of China under Grant No. 10471127, Zhejiang Provincial Natural Science Foundation of China under Grant No. R604001, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
Rights and permissions
About this article
Cite this article
Zhang, X., Ge, G. Existence of Z-cyclic 3PDTWh(p) for Prime p ≡ 1 (mod 4). Des. Codes Cryptogr. 45, 139–155 (2007). https://doi.org/10.1007/s10623-007-9103-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-007-9103-4