Existence of Z-cyclic 3PTWh(p) for any prime p ≡ 1 (mod 4)

1. Existence of Z-cyclic 3PTWh(p) for any prime p ≡ 1 (mod 4) 冯 弢(Tao Feng) 常彦勋(Yanxun Chang) Beijing Jiaotong University

2. a parter d b c parter Triplewhist tournament （TWh） • Let X be a set of vplayers,v = 4n (or 4n+1). LetB be a collection of ordered 4-subsets (a, b, c, d) of X(calledgames), where the unordered pairs {a, c}, {b, d} are calledparters, the pairs {a, b}, {c, d}opponents of the first kind,{a, d}, {b, c}opponents of the second kind.

3. a a d b d b c c Opponent of the second kind Opponent of the first kind Triplewhist tournament （TWh） • Let X be a set of vplayers,v = 4n (or 4n+1). LetB be a collection of ordered 4-subsets (a, b, c, d) of X(calledgames), where the unordered pairs {a, c}, {b, d} are calledparters, the pairs {a, b}, {c, d}opponents of the first kind,{a, d}, {b, c}opponents of the second kind.

4. Triplewhist tournament （TWh） • the games are arranged into 4n-1 (or 4n+1) rounds, each of n games • each player plays in exactly one game in each round (or all rounds but one) • each player partners every other player exactly once • each player has every other player as an opponent of the first kind exactly once, and that of the second kind exactly once. • TWh(4)

5. Z-cyclic TWh(4) m= v, A =if v≡ 1 (mod 4) m = v - 1, A = {∞}if v≡ 0 (mod 4) Z-cyclic Triplewhist tournament （ Z-cyclic TWh） • A triplewhist tournament is said to beZ-cyclicif • the players are elements in Zm∪A, where • the round j+1 is obtained by adding 1 (mod m) to every element in round j, where ∞ + 1 = ∞.

6. Z-cyclic TWh(4) Z-cyclic Triplewhist tournament with three-person property (Z-cyclic 3PTWh) • AZ-cyclic triplewhist tournament is said to havethree-person propertyif the intersection of any two games in the tournament is at most two.

7. Main Result TheoremThere exists a Z-cyclic 3PTWh(p) for any prime p ≡ 1 (mod 4) with the only exceptions of p=5, 13, 17. Z-cyclic 3PTWh(p) withp a prime

8. Lemma[Buratti, 2000] • Letp ≡ 5 (mod 8) be a primeand let (a, b, c, d) be a • quadruple of elements of Zp satisfying the following • conditions: • {a, b, c, d} is a representative system of the coset • classes , , , }; • (2)Each of the sets {a-b, c-d}, {a-c, b-d}, {a-d, b-c} • is a representative system of the coset classes { , }. • ThenR = {(ay, by, cy, dy)∣y ∈ }is the initial round • of a Z-cyclic TWh(p).

9. Let G be an abelian group, and a, b, c are pairwise distinct elements of G. • LetO(a, b, c) = {{a+g, b+g, c+g}: g ∈G},which is called theorbitof {a, b, c} under G. • If the order of G is a prime p, p ≠ 3, then ︱O(a, b, c)︱= p. • O(a, b, c)?O(a’, b’, c’) • Let G(a, b, c)={{b-a, c-a}, {a-b, c-b}, {a-c, b-c}}, which is calledthe generating setfor O(a, b, c) O(a, b, c) ∩ O(a’, b’, c’) ≠ , then G(a, b, c) = G(a’, b’, c’) O(a, b, c) = O(a’, b’, c’) iff G(a, b, c) = G(a’, b’, c’)

10. Lemma[T. Feng, Y. Chang, 2006] • Letp ≡ 5 (mod 8) be a primeand let (a, b, c, d) be a • quadruple of elements of Zp satisfying the following • conditions: • {a, b, c, d} is a representative system of the coset • classes , , , }; • (2)b-a ∈, c-a ∈ , c-b ∈ , • d-a ∈ , d-b∈ , d-c∈ , • ThenR = {(ay, by, cy, dy)∣y ∈ }is the initial round • of a Z-cyclic 3PTWh(p).

11. Lemma [Y. Chang, L. Ji, 2004] Use Weil’s theorem to guarantee the existence of certain elements in Zp

12. References: • M. Buratti, Existence of Z-cyclic triplewhist tournaments for a prime number of players, J. Combin. Theory Ser.A 90 (2000), 315--325. • Y. Chang, L. Ji, Optimal (4up, 5, 1) Optical orthogonal codes, J. Combin. Des. 5 (2004), 346-361. • 3. T. Feng and Y. Chang, Existence of Z-cyclic 3PTWh(p) for any prime p ≡ 1 (mod 4), Des. Codes Crypt. 39 (2006), 39-49.

13. Thank you