MinT - Best Known (216−97, 216, s)-Nets in Base 3

Best Known (216−97, 216, s)-Nets in Base 3

(216−97, 216, 84)-Net over F₃ — Constructive and digital

Digital (119, 216, 84)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 74, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]
2. digital (45, 142, 48)-net over F₃, using
  - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
      - an explicitly constructive algebraic function field [i]

(216−97, 216, 145)-Net over F₃ — Digital

Digital (119, 216, 145)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−97, 216, 1238)-Net in Base 3 — Upper bound on s

There is no (119, 216, 1239)-net in base 3, because

1 times m-reduction [i] would yield (119, 215, 1239)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 940689 813265 017296 005749 377492 869803 403548 926055 905154 173875 511571 646756 394570 890326 486470 004197 285025 > 3²¹⁵ [i]