MinT - Best Known (247−113, 247, s)-Nets in Base 3

Best Known (247−113, 247, s)-Nets in Base 3

(247−113, 247, 86)-Net over F₃ — Constructive and digital

Digital (134, 247, 86)-net over F₃, using

1 times m-reduction [i] based on digital (134, 248, 86)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (32, 89, 38)-net over F₃, using
    - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, 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]

(247−113, 247, 155)-Net over F₃ — Digital

Digital (134, 247, 155)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−113, 247, 1299)-Net in Base 3 — Upper bound on s

There is no (134, 247, 1300)-net in base 3, because

1 times m-reduction [i] would yield (134, 246, 1300)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2436 354293 107867 548076 006530 175308 287474 781752 608545 693320 581242 129841 024510 921137 073077 083576 216563 458347 216160 315201 > 3²⁴⁶ [i]