MinT - Best Known (229−105, 229, s)-Nets in Base 3

Best Known (229−105, 229, s)-Nets in Base 3

(229−105, 229, 85)-Net over F₃ — Constructive and digital

Digital (124, 229, 85)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 79, 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, 150, 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]

(229−105, 229, 144)-Net over F₃ — Digital

Digital (124, 229, 144)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−105, 229, 1199)-Net in Base 3 — Upper bound on s

There is no (124, 229, 1200)-net in base 3, because

1 times m-reduction [i] would yield (124, 228, 1200)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 276297 919557 448396 597544 116756 291311 682350 661717 978305 365082 773765 380782 294001 857848 154068 104900 784907 552897 > 3²²⁸ [i]