MinT - Best Known (105, 105+105, s)-Nets in Base 3

Best Known (105, 105+105, s)-Nets in Base 3

(105, 105+105, 73)-Net over F₃ — Constructive and digital

Digital (105, 210, 73)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 78, 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]
2. digital (27, 132, 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]

(105, 105+105, 104)-Net over F₃ — Digital

Digital (105, 210, 104)-net over F₃, using

t-expansion [i] based on digital (102, 210, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(105, 105+105, 786)-Net in Base 3 — Upper bound on s

There is no (105, 210, 787)-net in base 3, because

1 times m-reduction [i] would yield (105, 209, 787)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5427 877487 915073 835369 807217 461795 432728 889022 616514 528926 659546 803728 734060 980119 338109 010859 782521 > 3²⁰⁹ [i]