MinT - Best Known (99, 99+101, s)-Nets in Base 3

Best Known (99, 99+101, s)-Nets in Base 3

(99, 99+101, 69)-Net over F₃ — Constructive and digital

Digital (99, 200, 69)-net over F₃, using

1 times m-reduction [i] based on digital (99, 201, 69)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 72, 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, 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]

(99, 99+101, 97)-Net over F₃ — Digital

Digital (99, 200, 97)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 99+101, 723)-Net in Base 3 — Upper bound on s

There is no (99, 200, 724)-net in base 3, because

1 times m-reduction [i] would yield (99, 199, 724)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 90266 713998 185768 064678 469443 986935 681499 545212 540450 912158 761779 432157 127035 262843 504234 703897 > 3¹⁹⁹ [i]