MinT - Best Known (188−87, 188, s)-Nets in Base 3

Best Known (188−87, 188, s)-Nets in Base 3

(188−87, 188, 74)-Net over F₃ — Constructive and digital

Digital (101, 188, 74)-net over F₃, using

7 times m-reduction [i] based on digital (101, 195, 74)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 74, 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, 121, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃ (see above)

(188−87, 188, 118)-Net over F₃ — Digital

Digital (101, 188, 118)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−87, 188, 961)-Net in Base 3 — Upper bound on s

There is no (101, 188, 962)-net in base 3, because

1 times m-reduction [i] would yield (101, 187, 962)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 171137 003132 429134 614701 570982 990318 335083 450302 088409 136131 144541 006678 253433 938764 483649 > 3¹⁸⁷ [i]