MinT - Best Known (146, 146+69, s)-Nets in Base 3

Best Known (146, 146+69, s)-Nets in Base 3

(146, 146+69, 167)-Net over F₃ — Constructive and digital

Digital (146, 215, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 43, 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 (103, 172, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 86, 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]

(146, 146+69, 382)-Net over F₃ — Digital

Digital (146, 215, 382)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(146, 146+69, 6782)-Net in Base 3 — Upper bound on s

There is no (146, 215, 6783)-net in base 3, because

1 times m-reduction [i] would yield (146, 214, 6783)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 275838 156833 200252 999032 087555 714358 879851 049032 778176 073208 616490 541737 801736 084611 002173 535882 147005 > 3²¹⁴ [i]