MinT - Best Known (190−65, 190, s)-Nets in Base 3

Best Known (190−65, 190, s)-Nets in Base 3

(190−65, 190, 156)-Net over F₃ — Constructive and digital

Digital (125, 190, 156)-net over F₃, using

16 times m-reduction [i] based on digital (125, 206, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 103, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(190−65, 190, 281)-Net over F₃ — Digital

Digital (125, 190, 281)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(190−65, 190, 4174)-Net in Base 3 — Upper bound on s

There is no (125, 190, 4175)-net in base 3, because

1 times m-reduction [i] would yield (125, 189, 4175)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 506372 637389 443361 505074 242258 624226 668901 454295 009946 051237 978579 771205 931476 581491 923905 > 3¹⁸⁹ [i]