MinT - Best Known (140, 140+83, s)-Nets in Base 3

Best Known (140, 140+83, s)-Nets in Base 3

(140, 140+83, 156)-Net over F₃ — Constructive and digital

Digital (140, 223, 156)-net over F₃, using

13 times m-reduction [i] based on digital (140, 236, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 118, 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]

(140, 140+83, 253)-Net over F₃ — Digital

Digital (140, 223, 253)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(140, 140+83, 3052)-Net in Base 3 — Upper bound on s

There is no (140, 223, 3053)-net in base 3, because

1 times m-reduction [i] would yield (140, 222, 3053)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8413 770043 619151 888768 699523 235006 441009 561437 202514 366441 035244 852806 989667 749624 458448 469480 250889 371963 > 3²²² [i]