MinT - Best Known (239−117, 239, s)-Nets in Base 3

Best Known (239−117, 239, s)-Nets in Base 3

(239−117, 239, 78)-Net over F₃ — Constructive and digital

Digital (122, 239, 78)-net over F₃, using

t-expansion [i] based on digital (121, 239, 78)-net over F₃, using
- net from sequence [i] based on digital (121, 77)-sequence over F₃, using
  - base reduction for sequences [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]

(239−117, 239, 124)-Net over F₃ — Digital

Digital (122, 239, 124)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−117, 239, 962)-Net in Base 3 — Upper bound on s

There is no (122, 239, 963)-net in base 3, because

1 times m-reduction [i] would yield (122, 238, 963)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 369254 030282 263565 900035 695483 015292 951129 276624 658881 991750 348865 263440 506152 528954 094964 576278 275856 950017 647349 > 3²³⁸ [i]