MinT - Best Known (204−99, 204, s)-Nets in Base 3

Best Known (204−99, 204, s)-Nets in Base 3

(204−99, 204, 74)-Net over F₃ — Constructive and digital

Digital (105, 204, 74)-net over F₃, using

3 times m-reduction [i] based on digital (105, 207, 74)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 78, 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]
  2. digital (27, 129, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃ (see above)

(204−99, 204, 111)-Net over F₃ — Digital

Digital (105, 204, 111)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(204−99, 204, 858)-Net in Base 3 — Upper bound on s

There is no (105, 204, 859)-net in base 3, because

1 times m-reduction [i] would yield (105, 203, 859)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 519361 429620 209421 556039 473493 868901 606065 858770 265795 685020 811612 908641 030939 692495 061712 015831 > 3²⁰³ [i]