MinT - Best Known (228−141, 228, s)-Nets in Base 3

Best Known (228−141, 228, s)-Nets in Base 3

(228−141, 228, 62)-Net over F₃ — Constructive and digital

Digital (87, 228, 62)-net over F₃, using

net from sequence [i] based on digital (87, 61)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 61)-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]

(228−141, 228, 84)-Net over F₃ — Digital

Digital (87, 228, 84)-net over F₃, using

t-expansion [i] based on digital (71, 228, 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]

(228−141, 228, 408)-Net in Base 3 — Upper bound on s

There is no (87, 228, 409)-net in base 3, because

1 times m-reduction [i] would yield (87, 227, 409)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 162010 600563 894679 455612 992699 619500 070653 647370 625857 352042 732475 254197 815695 978383 170869 303921 914584 715977 > 3²²⁷ [i]