MinT - Best Known (187−93, 187, s)-Nets in Base 3

Best Known (187−93, 187, s)-Nets in Base 3

(187−93, 187, 69)-Net over F₃ — Constructive and digital

Digital (94, 187, 69)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 67, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]
2. digital (27, 120, 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]

(187−93, 187, 96)-Net over F₃ — Digital

Digital (94, 187, 96)-net over F₃, using

t-expansion [i] based on digital (89, 187, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(187−93, 187, 720)-Net in Base 3 — Upper bound on s

There is no (94, 187, 721)-net in base 3, because

1 times m-reduction [i] would yield (94, 186, 721)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 58327 451585 047432 478194 493258 766570 550660 118416 662495 411774 653906 860651 060755 799322 108697 > 3¹⁸⁶ [i]