Z-cyclic Whist Tournaments

In [1], Leonard gives a process for constructing of Z-cyclic whist tournaments for q² players, where q is a prime congruent to 3 modulo 4. The construction seems to work for all such q > 7, but this is not proven. Here we provide data from computations for 7 < q < 5000 (range extended 11/06):

The file provides data for quickly constructing a single Z-cyclic whist tournament on q² players for each prime considered. It contains a list of the form

where each term of the list has the form

Here

• q is a prime congruent to 3 modulo 4;
• g is an element of the multiplicative group (Z/(q²))^× of order (q-1)/2;
• the pairs [A_i, B_i] provide the needed data to write down the tables of the tournament. The number of these is m = (q+1)/2.

Given this information, one constructs the initial round tables of the form

where all computations are done in Z/(q²). Here, 1 ≤ k ≤ (q+1)/2 and 0 ≤ j < (q-1)/2. Subsequent rounds of the whist tournament come successively adding 1 to each element of the previous round, computations again done in Z/(q²).

[1] Leonard, Philip A., Some new Z-cyclic whist tournaments, Utilitas Math. 49 (1996), 223-232.