MinT - Best Known (84−40, 84, s)-Nets in Base 3

Best Known (84−40, 84, s)-Nets in Base 3

(84−40, 84, 47)-Net over F₃ — Constructive and digital

Digital (44, 84, 47)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 29, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 55, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]

(84−40, 84, 57)-Net over F₃ — Digital

Digital (44, 84, 57)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(84−40, 84, 399)-Net in Base 3 — Upper bound on s

There is no (44, 84, 400)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 12062 671618 713267 344182 141972 768474 874241 > 3⁸⁴ [i]