MinT - Best Known (198−101, 198, s)-Nets in Base 3

Best Known (198−101, 198, s)-Nets in Base 3

(198−101, 198, 68)-Net over F₃ — Constructive and digital

Digital (97, 198, 68)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 71, 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, 127, 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]

(198−101, 198, 96)-Net over F₃ — Digital

Digital (97, 198, 96)-net over F₃, using

t-expansion [i] based on digital (89, 198, 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]

(198−101, 198, 690)-Net in Base 3 — Upper bound on s

There is no (97, 198, 691)-net in base 3, because

1 times m-reduction [i] would yield (97, 197, 691)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10128 465549 347591 705236 083008 080528 022816 723422 439751 205453 323281 233754 372120 782721 275530 028197 > 3¹⁹⁷ [i]