MinT - Best Known (219−97, 219, s)-Nets in Base 3

Best Known (219−97, 219, s)-Nets in Base 3

(219−97, 219, 85)-Net over F₃ — Constructive and digital

Digital (122, 219, 85)-net over F₃, using

3 times m-reduction [i] based on digital (122, 222, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 77, 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 (45, 145, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(219−97, 219, 152)-Net over F₃ — Digital

Digital (122, 219, 152)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−97, 219, 1329)-Net in Base 3 — Upper bound on s

There is no (122, 219, 1330)-net in base 3, because

1 times m-reduction [i] would yield (122, 218, 1330)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 105 141101 823180 900085 976302 319030 135756 926579 035781 946295 923719 833980 205641 915487 000358 213162 469063 557089 > 3²¹⁸ [i]