MinT - Best Known (196−99, 196, s)-Nets in Base 3

Best Known (196−99, 196, s)-Nets in Base 3

(196−99, 196, 69)-Net over F₃ — Constructive and digital

Digital (97, 196, 69)-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 (27, 126, 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]

(196−99, 196, 96)-Net over F₃ — Digital

Digital (97, 196, 96)-net over F₃, using

t-expansion [i] based on digital (89, 196, 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]

(196−99, 196, 709)-Net in Base 3 — Upper bound on s

There is no (97, 196, 710)-net in base 3, because

1 times m-reduction [i] would yield (97, 195, 710)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1119 563709 233508 050754 359087 581622 034331 495372 558159 141163 601245 171883 567889 301359 815639 906093 > 3¹⁹⁵ [i]