MinT - Best Known (218−128, 218, s)-Nets in Base 3

Best Known (218−128, 218, s)-Nets in Base 3

(218−128, 218, 64)-Net over F₃ — Constructive and digital

Digital (90, 218, 64)-net over F₃, using

t-expansion [i] based on digital (89, 218, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(218−128, 218, 96)-Net over F₃ — Digital

Digital (90, 218, 96)-net over F₃, using

t-expansion [i] based on digital (89, 218, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(218−128, 218, 460)-Net in Base 3 — Upper bound on s

There is no (90, 218, 461)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 116 317078 824462 367257 557209 565664 658522 528358 235393 592689 111911 138923 478638 083306 260004 800119 992652 867329 > 3²¹⁸ [i]