MinT - Best Known (78, 78+111, s)-Nets in Base 3

Best Known (78, 78+111, s)-Nets in Base 3

(78, 78+111, 53)-Net over F₃ — Constructive and digital

Digital (78, 189, 53)-net over F₃, using

net from sequence [i] based on digital (78, 52)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 52)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(78, 78+111, 84)-Net over F₃ — Digital

Digital (78, 189, 84)-net over F₃, using

t-expansion [i] based on digital (71, 189, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(78, 78+111, 404)-Net in Base 3 — Upper bound on s

There is no (78, 189, 405)-net in base 3, because

1 times m-reduction [i] would yield (78, 188, 405)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 557247 369413 281430 283370 471209 214392 517899 984688 767189 510950 952149 126213 440381 447198 565083 > 3¹⁸⁸ [i]