MinT - Best Known (176−87, 176, s)-Nets in Base 3

Best Known (176−87, 176, s)-Nets in Base 3

(176−87, 176, 65)-Net over F₃ — Constructive and digital

Digital (89, 176, 65)-net over F₃, using

7 times m-reduction [i] based on digital (89, 183, 65)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 62, 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 (27, 121, 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]

(176−87, 176, 96)-Net over F₃ — Digital

Digital (89, 176, 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]

(176−87, 176, 696)-Net in Base 3 — Upper bound on s

There is no (89, 176, 697)-net in base 3, because

1 times m-reduction [i] would yield (89, 175, 697)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 318044 772420 255266 511418 905004 552470 345381 870441 501760 625922 704187 004995 154238 522907 > 3¹⁷⁵ [i]