MinT - Best Known (197−99, 197, s)-Nets in Base 3

Best Known (197−99, 197, s)-Nets in Base 3

(197−99, 197, 69)-Net over F₃ — Constructive and digital

Digital (98, 197, 69)-net over F₃, using

1 times m-reduction [i] based on digital (98, 198, 69)-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 (27, 127, 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]

(197−99, 197, 97)-Net over F₃ — Digital

Digital (98, 197, 97)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(197−99, 197, 726)-Net in Base 3 — Upper bound on s

There is no (98, 197, 727)-net in base 3, because

1 times m-reduction [i] would yield (98, 196, 727)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3328 399827 930590 272247 197611 405305 093319 868616 840851 815585 163792 352001 008938 225975 768954 158927 > 3¹⁹⁶ [i]