MinT - Best Known (195−99, 195, s)-Nets in Base 3

Best Known (195−99, 195, s)-Nets in Base 3

(195−99, 195, 68)-Net over F₃ — Constructive and digital

Digital (96, 195, 68)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 70, 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 (26, 125, 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]

(195−99, 195, 96)-Net over F₃ — Digital

Digital (96, 195, 96)-net over F₃, using

t-expansion [i] based on digital (89, 195, 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]

(195−99, 195, 692)-Net in Base 3 — Upper bound on s

There is no (96, 195, 693)-net in base 3, because

1 times m-reduction [i] would yield (96, 194, 693)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 367 338698 387721 006222 859437 457880 685606 463393 745631 296082 121592 932255 884006 825241 144504 800939 > 3¹⁹⁴ [i]