MinT - Best Known (246−115, 246, s)-Nets in Base 3

Best Known (246−115, 246, s)-Nets in Base 3

(246−115, 246, 85)-Net over F₃ — Constructive and digital

Digital (131, 246, 85)-net over F₃, using

3 times m-reduction [i] based on digital (131, 249, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 86, 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, 163, 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]

(246−115, 246, 145)-Net over F₃ — Digital

Digital (131, 246, 145)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(246−115, 246, 1185)-Net in Base 3 — Upper bound on s

There is no (131, 246, 1186)-net in base 3, because

1 times m-reduction [i] would yield (131, 245, 1186)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 803 550346 231613 150032 522084 164099 619555 834174 594407 668988 103877 422036 053610 356291 357335 102843 397485 114906 411769 367701 > 3²⁴⁵ [i]