MinT - Best Known (138−129, 138, s)-Nets in Base 3

Best Known (138−129, 138, s)-Nets in Base 3

(138−129, 138, 19)-Net over F₃ — Constructive and digital

Digital (9, 138, 19)-net over F₃, using

net from sequence [i] based on digital (9, 18)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
  - an explicitly constructive algebraic function field [i]

(138−129, 138, 26)-Net in Base 3 — Upper bound on s

There is no (9, 138, 27)-net in base 3, because

61 times m-reduction [i] would yield (9, 77, 27)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3⁷⁷, 27, S₃, 3, 68), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 147 808829 414345 923316 083210 206383 297601 / 23 > 3⁷⁷ [i]