MinT - Best Known (189−91, 189, s)-Nets in Base 3

Best Known (189−91, 189, s)-Nets in Base 3

(189−91, 189, 73)-Net over F₃ — Constructive and digital

Digital (98, 189, 73)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 71, 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, 118, 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]

(189−91, 189, 106)-Net over F₃ — Digital

Digital (98, 189, 106)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−91, 189, 824)-Net in Base 3 — Upper bound on s

There is no (98, 189, 825)-net in base 3, because

1 times m-reduction [i] would yield (98, 188, 825)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 518639 743940 644305 258788 053553 334950 193398 087647 226000 840217 907186 124148 897149 924108 155587 > 3¹⁸⁸ [i]