MinT - Best Known (99, 99+85, s)-Nets in Base 3

Best Known (99, 99+85, s)-Nets in Base 3

(99, 99+85, 74)-Net over F₃ — Constructive and digital

Digital (99, 184, 74)-net over F₃, using

5 times m-reduction [i] based on digital (99, 189, 74)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 72, 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, 117, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃ (see above)

(99, 99+85, 116)-Net over F₃ — Digital

Digital (99, 184, 116)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 99+85, 949)-Net in Base 3 — Upper bound on s

There is no (99, 184, 950)-net in base 3, because

1 times m-reduction [i] would yield (99, 183, 950)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2122 398999 921879 061564 045104 888341 459142 260662 887614 333151 573023 637739 958787 517828 070877 > 3¹⁸³ [i]