MinT - Best Known (239−110, 239, s)-Nets in Base 3

Best Known (239−110, 239, s)-Nets in Base 3

(239−110, 239, 85)-Net over F₃ — Constructive and digital

Digital (129, 239, 85)-net over F₃, using

4 times m-reduction [i] based on digital (129, 243, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 84, 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, 159, 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]

(239−110, 239, 148)-Net over F₃ — Digital

Digital (129, 239, 148)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−110, 239, 1209)-Net in Base 3 — Upper bound on s

There is no (129, 239, 1210)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 097514 967803 761207 923305 929463 599341 076583 555241 120784 036605 706802 622443 563125 827553 583464 882759 624680 075812 673705 > 3²³⁹ [i]