MinT - Best Known (169−53, 169, s)-Nets in Base 3

Best Known (169−53, 169, s)-Nets in Base 3

(169−53, 169, 158)-Net over F₃ — Constructive and digital

Digital (116, 169, 158)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 29, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (87, 140, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 70, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(169−53, 169, 333)-Net over F₃ — Digital

Digital (116, 169, 333)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−53, 169, 6360)-Net in Base 3 — Upper bound on s

There is no (116, 169, 6361)-net in base 3, because

1 times m-reduction [i] would yield (116, 168, 6361)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 143 853883 623169 068157 915203 860277 217071 199134 657558 415960 565233 968213 561239 329113 > 3¹⁶⁸ [i]