MinT - Best Known (58, 111, s)-Nets in Base 3

Best Known (58, 111, s)-Nets in Base 3

(58, 111, 56)-Net over F₃ — Constructive and digital

Digital (58, 111, 56)-net over F₃, using

3 times m-reduction [i] based on digital (58, 114, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 43, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (15, 71, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(58, 111, 70)-Net over F₃ — Digital

Digital (58, 111, 70)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 111, 525)-Net in Base 3 — Upper bound on s

There is no (58, 111, 526)-net in base 3, because

1 times m-reduction [i] would yield (58, 110, 526)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 31009 139835 135826 074768 681693 720318 718687 291795 912845 > 3¹¹⁰ [i]