MinT - Best Known (126, 126+104, s)-Nets in Base 3

Best Known (126, 126+104, s)-Nets in Base 3

(126, 126+104, 85)-Net over F₃ — Constructive and digital

Digital (126, 230, 85)-net over F₃, using

4 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, 126+104, 150)-Net over F₃ — Digital

Digital (126, 230, 150)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+104, 1253)-Net in Base 3 — Upper bound on s

There is no (126, 230, 1254)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 56 617315 005718 974413 872310 946021 867680 214062 708715 419958 142238 763890 227558 333509 424791 735388 825228 793668 130793 > 3²³⁰ [i]