MinT - Best Known (178−93, 178, s)-Nets in Base 3

Best Known (178−93, 178, s)-Nets in Base 3

(178−93, 178, 60)-Net over F₃ — Constructive and digital

Digital (85, 178, 60)-net over F₃, using

net from sequence [i] based on digital (85, 59)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 59)-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]

(178−93, 178, 84)-Net over F₃ — Digital

Digital (85, 178, 84)-net over F₃, using

t-expansion [i] based on digital (71, 178, 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]

(178−93, 178, 572)-Net in Base 3 — Upper bound on s

There is no (85, 178, 573)-net in base 3, because

1 times m-reduction [i] would yield (85, 177, 573)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 917102 428986 360252 844502 369839 327372 804485 786434 487671 664106 473041 756718 595238 932369 > 3¹⁷⁷ [i]