MinT - Best Known (78, 78+119, s)-Nets in Base 3

Best Known (78, 78+119, s)-Nets in Base 3

(78, 78+119, 53)-Net over F₃ — Constructive and digital

Digital (78, 197, 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+119, 84)-Net over F₃ — Digital

Digital (78, 197, 84)-net over F₃, using

t-expansion [i] based on digital (71, 197, 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+119, 383)-Net in Base 3 — Upper bound on s

There is no (78, 197, 384)-net in base 3, because

1 times m-reduction [i] would yield (78, 196, 384)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3562 702828 796437 874241 106685 666368 933889 924586 931247 539393 476560 388118 124807 419370 193838 933505 > 3¹⁹⁶ [i]