MinT - Best Known (42, 42+37, s)-Nets in Base 3

Best Known (42, 42+37, s)-Nets in Base 3

(42, 42+37, 47)-Net over F₃ — Constructive and digital

Digital (42, 79, 47)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 27, 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, 52, 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]

(42, 42+37, 57)-Net over F₃ — Digital

Digital (42, 79, 57)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+37, 423)-Net in Base 3 — Upper bound on s

There is no (42, 79, 424)-net in base 3, because

1 times m-reduction [i] would yield (42, 78, 424)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16 440399 083572 458438 709237 924773 854065 > 3⁷⁸ [i]