MinT - Best Known (230−105, 230, s)-Nets in Base 3

Best Known (230−105, 230, s)-Nets in Base 3

(230−105, 230, 85)-Net over F₃ — Constructive and digital

Digital (125, 230, 85)-net over F₃, using

1 times m-reduction [i] based on digital (125, 231, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 80, 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, 151, 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]

(230−105, 230, 146)-Net over F₃ — Digital

Digital (125, 230, 146)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−105, 230, 1225)-Net in Base 3 — Upper bound on s

There is no (125, 230, 1226)-net in base 3, because

1 times m-reduction [i] would yield (125, 229, 1226)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 310033 035703 310426 585334 828449 971548 669083 179520 707713 389645 495895 262171 853649 546848 138999 118160 539310 423689 > 3²²⁹ [i]