MinT - Best Known (165−75, 165, s)-Nets in Base 3

Best Known (165−75, 165, s)-Nets in Base 3

(165−75, 165, 73)-Net over F₃ — Constructive and digital

Digital (90, 165, 73)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 63, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]
2. digital (27, 102, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(165−75, 165, 111)-Net over F₃ — Digital

Digital (90, 165, 111)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−75, 165, 918)-Net in Base 3 — Upper bound on s

There is no (90, 165, 919)-net in base 3, because

1 times m-reduction [i] would yield (90, 164, 919)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 812643 445214 354541 095087 179333 458848 920642 375964 990704 963166 232608 601667 811495 > 3¹⁶⁴ [i]