MinT - Best Known (227−81, 227, s)-Nets in Base 3

Best Known (227−81, 227, s)-Nets in Base 3

(227−81, 227, 162)-Net over F₃ — Constructive and digital

Digital (146, 227, 162)-net over F₃, using

1 times m-reduction [i] based on digital (146, 228, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 114, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(227−81, 227, 290)-Net over F₃ — Digital

Digital (146, 227, 290)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−81, 227, 3873)-Net in Base 3 — Upper bound on s

There is no (146, 227, 3874)-net in base 3, because

1 times m-reduction [i] would yield (146, 226, 3874)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 678318 055613 833756 883760 913440 044757 341433 295134 074129 362164 017319 249255 430632 052431 946033 949702 150410 670481 > 3²²⁶ [i]