MinT - Best Known (188−89, 188, s)-Nets in Base 3

Best Known (188−89, 188, s)-Nets in Base 3

(188−89, 188, 74)-Net over F₃ — Constructive and digital

Digital (99, 188, 74)-net over F₃, using

1 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)

(188−89, 188, 111)-Net over F₃ — Digital

Digital (99, 188, 111)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−89, 188, 877)-Net in Base 3 — Upper bound on s

There is no (99, 188, 878)-net in base 3, because

1 times m-reduction [i] would yield (99, 187, 878)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 174251 072892 959650 028789 793195 976988 943631 816319 939080 807312 570073 755223 155798 899896 897049 > 3¹⁸⁷ [i]