MinT - Best Known (158−55, 158, s)-Nets in Base 3

Best Known (158−55, 158, s)-Nets in Base 3

(158−55, 158, 156)-Net over F₃ — Constructive and digital

Digital (103, 158, 156)-net over F₃, using

4 times m-reduction [i] based on digital (103, 162, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 81, 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]

(158−55, 158, 229)-Net over F₃ — Digital

Digital (103, 158, 229)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(158−55, 158, 3222)-Net in Base 3 — Upper bound on s

There is no (103, 158, 3223)-net in base 3, because

1 times m-reduction [i] would yield (103, 157, 3223)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 811 987156 361089 795466 252351 140186 926644 627039 553504 411439 159046 123153 435563 > 3¹⁵⁷ [i]