MinT - Best Known (165−83, 165, s)-Nets in Base 3

Best Known (165−83, 165, s)-Nets in Base 3

(165−83, 165, 64)-Net over F₃ — Constructive and digital

Digital (82, 165, 64)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 56, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (26, 109, 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]

(165−83, 165, 84)-Net over F₃ — Digital

Digital (82, 165, 84)-net over F₃, using

t-expansion [i] based on digital (71, 165, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(165−83, 165, 614)-Net in Base 3 — Upper bound on s

There is no (82, 165, 615)-net in base 3, because

1 times m-reduction [i] would yield (82, 164, 615)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 861170 098429 767204 112394 997640 974648 715029 564975 619750 674265 288103 571924 204671 > 3¹⁶⁴ [i]