MinT - Best Known (176−99, 176, s)-Nets in Base 3

Best Known (176−99, 176, s)-Nets in Base 3

(176−99, 176, 52)-Net over F₃ — Constructive and digital

Digital (77, 176, 52)-net over F₃, using

net from sequence [i] based on digital (77, 51)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 51)-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]

(176−99, 176, 84)-Net over F₃ — Digital

Digital (77, 176, 84)-net over F₃, using

t-expansion [i] based on digital (71, 176, 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]

(176−99, 176, 436)-Net in Base 3 — Upper bound on s

There is no (77, 176, 437)-net in base 3, because

1 times m-reduction [i] would yield (77, 175, 437)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 320021 536745 877031 824473 115937 984956 868052 499226 369924 287833 793738 806860 529273 533611 > 3¹⁷⁵ [i]