MinT - Best Known (93, 93+99, s)-Nets in Base 3

Best Known (93, 93+99, s)-Nets in Base 3

(93, 93+99, 65)-Net over F₃ — Constructive and digital

Digital (93, 192, 65)-net over F₃, using

3 times m-reduction [i] based on digital (93, 195, 65)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 66, 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, 129, 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]

(93, 93+99, 96)-Net over F₃ — Digital

Digital (93, 192, 96)-net over F₃, using

t-expansion [i] based on digital (89, 192, 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]

(93, 93+99, 644)-Net in Base 3 — Upper bound on s

There is no (93, 192, 645)-net in base 3, because

1 times m-reduction [i] would yield (93, 191, 645)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13 668482 016337 250826 543267 177989 181057 729786 378879 300636 452535 250064 835637 953346 131147 244619 > 3¹⁹¹ [i]