MinT - Best Known (78, 194, s)-Nets in Base 3

Best Known (78, 194, s)-Nets in Base 3

(78, 194, 53)-Net over F₃ — Constructive and digital

Digital (78, 194, 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, 194, 84)-Net over F₃ — Digital

Digital (78, 194, 84)-net over F₃, using

t-expansion [i] based on digital (71, 194, 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, 194, 388)-Net in Base 3 — Upper bound on s

There is no (78, 194, 389)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 411 125298 442244 647199 852586 252914 092793 219962 134080 103365 683858 213689 191011 658691 784022 009265 > 3¹⁹⁴ [i]