MinT - Best Known (240−89, 240, s)-Nets in Base 3

Best Known (240−89, 240, s)-Nets in Base 3

(240−89, 240, 156)-Net over F₃ — Constructive and digital

Digital (151, 240, 156)-net over F₃, using

t-expansion [i] based on digital (147, 240, 156)-net over F₃, using
- 10 times m-reduction [i] based on digital (147, 250, 156)-net over F₃, using
  - trace code for nets [i] based on digital (22, 125, 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]

(240−89, 240, 273)-Net over F₃ — Digital

Digital (151, 240, 273)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−89, 240, 3325)-Net in Base 3 — Upper bound on s

There is no (151, 240, 3326)-net in base 3, because

1 times m-reduction [i] would yield (151, 239, 3326)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 077160 818063 754179 564564 285746 503378 576884 785009 386593 946203 195879 146626 165327 040383 522784 453400 198044 819008 368153 > 3²³⁹ [i]