MinT - Best Known (230−45, 230, s)-Nets in Base 3

Best Known (230−45, 230, s)-Nets in Base 3

(230−45, 230, 692)-Net over F₃ — Constructive and digital

Digital (185, 230, 692)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 4)-net over F₃, using
  - net from sequence [i] based on digital (0, 3)-sequence over F₃, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 0 and N(F) ≥ 4, using
      - the rational function field F₃(x) [i]
    - Niederreiter sequence [i]
2. digital (163, 208, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 52, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(230−45, 230, 2713)-Net over F₃ — Digital

Digital (185, 230, 2713)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−45, 230, 418985)-Net in Base 3 — Upper bound on s

There is no (185, 230, 418986)-net in base 3, because

1 times m-reduction [i] would yield (185, 229, 418986)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 229419 043300 695493 654233 558521 180568 788948 268318 387849 800508 615760 023883 965775 386840 517073 613608 439048 687181 > 3²²⁹ [i]