MinT - Best Known (239−107, 239, s)-Nets in Base 3

Best Known (239−107, 239, s)-Nets in Base 3

(239−107, 239, 86)-Net over F₃ — Constructive and digital

Digital (132, 239, 86)-net over F₃, using

3 times m-reduction [i] based on digital (132, 242, 86)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (32, 87, 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, 155, 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]

(239−107, 239, 159)-Net over F₃ — Digital

Digital (132, 239, 159)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−107, 239, 1378)-Net in Base 3 — Upper bound on s

There is no (132, 239, 1379)-net in base 3, because

1 times m-reduction [i] would yield (132, 238, 1379)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 369777 015724 955774 607380 018905 901375 526636 223814 862319 137287 739621 540197 236530 997623 704731 226789 440579 704550 048095 > 3²³⁸ [i]