Z-Cyclic DTWh(p)/OTWh(p), for primes p = 2~k + 1 (mod 2~(k+1)), k = 5,6, 7 - An Empirical Study

摘要

Let p denote a prime such that p = 2~(k + 1) (mod 2~(k+1)), k ≥ 2. For k = 2, 3,4 it is known that Z-cyclic directed triplewhist designs and Z-cyclic ordered triplewhist designs exist for all such primes except for the impossible cases p = 5,13,17. Here, primes corresponding to k = 5,6, 7 are considered. Due to the nature of the constructions employed, the analytical asymptotic bounds obtained via applications of Weil's Theorem are so large that an attempt to establish complete existence for these cases is impractical. It is shown that for k = 5, 6, 7 and for all primes p = 2~(k + 1) (mod 2~(k+1)), p < 3,200,000, Z-cyclic directed triplewhist designs exist. For the same sets of primes it is established that Z-cyclic ordered triplewhist designs exist except, possibly, for p = 97,193,449,577,641,1409. Furthermore, for each k = 5,6,7, an empirical asymptotic bound, N_k, is introduced and it is conjectured that Z-cyclic directed triplewhist designs and Z-cyclic ordered triplewhist designs exist for all relevant primes greater than N_k. The conjectures are that N_5 = 84, 449, N_6 = 320, 833 and N_7 = 1,344,641.