MinT - Best Known (126, 227, s)-Nets in Base 3

Best Known (126, 227, s)-Nets in Base 3

(126, 227, 85)-Net over F₃ — Constructive and digital

Digital (126, 227, 85)-net over F₃, using

7 times m-reduction [i] based on digital (126, 234, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 81, 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, 153, 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]

(126, 227, 155)-Net over F₃ — Digital

Digital (126, 227, 155)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 227, 1348)-Net in Base 3 — Upper bound on s

There is no (126, 227, 1349)-net in base 3, because

1 times m-reduction [i] would yield (126, 226, 1349)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 692482 516907 068886 528408 783811 451177 606229 076528 326912 882173 490941 364197 839432 683195 250145 151122 423247 735697 > 3²²⁶ [i]