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In the past few years, several studies have appeared that relate to the existence of \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments. In these studies, the number of players in the tournament is taken to be a prime \( p \) of the form \( p \equiv 2^k + 1 \pmod{2^k+1} \), where \( k \geq 2 \). For the cases \( k = 2, 3, 4 \) it has been shown [6,4,5,12] that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all such primes except for the impossible cases \( p = 5, 13, 17 \). For the cases \( k = 5, 6, 7 \) it has been shown [13] that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments exist for all such primes less than \( 3{,}200{,}000 \) and that \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all such primes less than \( 3{,}200{,}000 \) with the exception that existence or non-existence of these designs for \( p = 97, 193, 449, 577, 641, 1409 \) is an open question. Here the case \( k = 8 \) is considered. It is established that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all primes \( p \equiv 257 \pmod{512} \), \( p \leq 6{,}944{,}177 \), except possibly for \( p = 257, 769, 3329 \). For \( p = 3329 \) we are able to construct a \( \mathbb{Z} \)-cyclic directed-triplewhist tournament, but the existence of a \( \mathbb{Z} \)-cyclic ordered-triplewhist tournament remains an open question. Furthermore, for each type of design it is conjectured that our basic constructions will produce these designs whenever \( p > 5{,}299{,}457 \).