MinT - Best Known (239−93, 239, s)-Nets in Base 3

Best Known (239−93, 239, s)-Nets in Base 3

(239−93, 239, 156)-Net over F₃ — Constructive and digital

Digital (146, 239, 156)-net over F₃, using

9 times m-reduction [i] based on digital (146, 248, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 124, 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]

(239−93, 239, 236)-Net over F₃ — Digital

Digital (146, 239, 236)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−93, 239, 2602)-Net in Base 3 — Upper bound on s

There is no (146, 239, 2603)-net in base 3, because

1 times m-reduction [i] would yield (146, 238, 2603)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 365065 626124 272249 215380 717833 386241 029169 204243 791157 047706 621530 443425 913270 213763 631075 918724 507650 141879 244317 > 3²³⁸ [i]