MinT - Best Known (239−97, 239, s)-Nets in Base 3

Best Known (239−97, 239, s)-Nets in Base 3

(239−97, 239, 156)-Net over F₃ — Constructive and digital

Digital (142, 239, 156)-net over F₃, using

1 times m-reduction [i] based on digital (142, 240, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 120, 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]

(239−97, 239, 210)-Net over F₃ — Digital

Digital (142, 239, 210)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−97, 239, 2128)-Net in Base 3 — Upper bound on s

There is no (142, 239, 2129)-net in base 3, because

1 times m-reduction [i] would yield (142, 238, 2129)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 366312 921228 463933 010113 832056 401126 624106 546574 824448 458470 281776 793400 950960 718882 654380 416220 489836 021557 569985 > 3²³⁸ [i]