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

Let X be a set of v players, v = 4n (or 4n+1). Let be a collection of ordered 4-subsets (a, b, c, d) of X (called games), where the unordered pairs {a, c}, {b, d} are called parters, the pairs {a, b}, {c, d} opponents of the first kind, {a, d}, {b, c} opponents of the second kind.

a

d b

c

parterparter

parterparter

Triplewhist tournament （ TWh ）

Let X be a set of v players, v = 4n (or 4n+1). Let be a collection of ordered 4-subsets (a, b, c, d) of X (called games), where the unordered pairs {a, c}, {b, d} are called parters, the pairs {a, b}, {c, d} opponents of the first kind, {a, d}, {b, c} opponents of the second kind.

a

d b

c

Opponent of the first kindOpponent of the first kind

a

d b

c

Opponent of the second kindOpponent of the second kind

Triplewhist tournament （ TWh ）

a) the games are arranged into 4n-1 (or 4n+1) rounds, each of n games

b) each player plays in exactly one game in each round (or all rounds but one)

c) each player partners every other player exactly once

d) each player has every other player as an opponent of the first kind exactly once, and that of the second kind exactly once.

Triplewhist tournament （ TWh ）

( ,1,0,2)

( , 2,1,0)

( ,0,2,1)

TWh(4)

Z-cyclic TWh(4)

( ,1,0,2)

( , 2,1,0)

( ,0,2,1)

Z-cyclic Triplewhist tournament （ Z-cyclic TWh ）

A triplewhist tournament is said to be Z-cyclic if

① the players are elements in Zm∪A, where

② the round j+1 is obtained by adding 1 (mod m) to every element in round j, where ∞ + 1 = ∞.

m = v, A = if v ≡ 1 (mod 4)

m = v - 1, A = {∞} if v ≡ 0 (mod 4)

A Z-cyclic triplewhist tournament is said to have three-person property if the intersection of any two games in the tournament is at most two.

Z-cyclic TWh(4)

( ,1,0,2)

( , 2,1,0)

( ,0, 2,1)

Z-cyclic Triplewhist tournament with three-person property (Z-cyclic 3PTWh)

Main Result

Theorem There 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) with p a prime

Lemma [Buratti, 2000]

Let p ≡ 5 (mod 8) be a prime and let (a, b, c, d) be aquadruple of elements of Zp satisfying the followingconditions:(1) {a, b, c, d} is a representative system of the cosetclasses , , , };(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 { , }.Then R = {(ay, by, cy, dy) y ∣ ∈ } is the initial round of a Z-cyclic TWh(p).

40C 4

1C 42C 4

3C

20C 2

1C40C

Let G be an abelian group, and a, b, c are pairwise distinct elements of G.

Let O(a, b, c) = {{a+g, b+g, c+g}: g ∈G}, which is called the orbit of {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

called the generating set for 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’)

Lemma [T. Feng, Y. Chang, 2006]

Let p ≡ 5 (mod 8) be a prime and let (a, b, c, d) be aquadruple of elements of Zp satisfying the followingconditions:(1) {a, b, c, d} is a representative system of the cosetclasses , , , };(2) b-a ∈ , c-a ∈ , c-b ∈ ,

d-a ∈ , d-b ∈ , d-c ∈ ,

Then R = {(ay, by, cy, dy) y ∣ ∈ } is the initial round of a Z-cyclic 3PTWh(p).

40C 4

1C 42C 4

3C

40C

40C 4

0C 42C

41C 4

1C 43C

Lemma [Y. Chang, L. Ji, 2004]

Use Weil’s theorem to guarantee the existence of certain elements in Zp

References:References:1. M. Buratti, Existence of Z-cyclic triplewhist

tournaments for a prime number of players, J. Combin. Theory Ser.A 90 (2000), 315--325.

2. 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.

Thank you