MinT - Best Known (212−73, 212, s)-Nets in Base 3

Best Known (212−73, 212, s)-Nets in Base 3

(212−73, 212, 162)-Net over F₃ — Constructive and digital

Digital (139, 212, 162)-net over F₃, using

2 times m-reduction [i] based on digital (139, 214, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 107, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(212−73, 212, 303)-Net over F₃ — Digital

Digital (139, 212, 303)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−73, 212, 4433)-Net in Base 3 — Upper bound on s

There is no (139, 212, 4434)-net in base 3, because

1 times m-reduction [i] would yield (139, 211, 4434)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47364 875909 664561 707573 837751 695859 981663 838607 579541 461498 569994 849362 006365 281544 682256 793289 179753 > 3²¹¹ [i]