MinT - Best Known (156−53, 156, s)-Nets in Base 3

Best Known (156−53, 156, s)-Nets in Base 3

(156−53, 156, 156)-Net over F₃ — Constructive and digital

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

6 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]

(156−53, 156, 244)-Net over F₃ — Digital

Digital (103, 156, 244)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(156−53, 156, 3661)-Net in Base 3 — Upper bound on s

There is no (103, 156, 3662)-net in base 3, because

1 times m-reduction [i] would yield (103, 155, 3662)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 90 358437 340479 151161 333102 850641 088389 869880 229179 738544 915940 736180 672525 > 3¹⁵⁵ [i]