MinT - Best Known (137, 137+75, s)-Nets in Base 3

Best Known (137, 137+75, s)-Nets in Base 3

(137, 137+75, 156)-Net over F₃ — Constructive and digital

Digital (137, 212, 156)-net over F₃, using

18 times m-reduction [i] based on digital (137, 230, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 115, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(137, 137+75, 280)-Net over F₃ — Digital

Digital (137, 212, 280)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+75, 3816)-Net in Base 3 — Upper bound on s

There is no (137, 212, 3817)-net in base 3, because

1 times m-reduction [i] would yield (137, 211, 3817)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47361 169619 787040 354144 967154 859353 971702 092138 973978 250603 011393 920997 650218 400570 018226 245338 181059 > 3²¹¹ [i]