MinT - Best Known (89, 89+42, s)-Nets in Base 3

Best Known (89, 89+42, s)-Nets in Base 3

(89, 89+42, 156)-Net over F₃ — Constructive and digital

Digital (89, 131, 156)-net over F₃, using

3 times m-reduction [i] based on digital (89, 134, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 67, 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]

(89, 89+42, 252)-Net over F₃ — Digital

Digital (89, 131, 252)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+42, 4089)-Net in Base 3 — Upper bound on s

There is no (89, 131, 4090)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 319 874079 555123 017919 908850 382916 859710 097208 467942 106365 287469 > 3¹³¹ [i]