MinT - Best Known (149, 149+71, s)-Nets in Base 3

Best Known (149, 149+71, s)-Nets in Base 3

(149, 149+71, 167)-Net over F₃ — Constructive and digital

Digital (149, 220, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 44, 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 (105, 176, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 88, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(149, 149+71, 382)-Net over F₃ — Digital

Digital (149, 220, 382)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(149, 149+71, 6689)-Net in Base 3 — Upper bound on s

There is no (149, 220, 6690)-net in base 3, because

1 times m-reduction [i] would yield (149, 219, 6690)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 308 852031 592064 195669 248704 872964 231553 647969 646339 647626 609596 893309 473356 269290 053609 649429 519009 137809 > 3²¹⁹ [i]