MinT - Best Known (115−55, 115, s)-Nets in Base 3

Best Known (115−55, 115, s)-Nets in Base 3

(115−55, 115, 56)-Net over F₃ — Constructive and digital

Digital (60, 115, 56)-net over F₃, using

5 times m-reduction [i] based on digital (60, 120, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 45, 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, 75, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(115−55, 115, 71)-Net over F₃ — Digital

Digital (60, 115, 71)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(115−55, 115, 538)-Net in Base 3 — Upper bound on s

There is no (60, 115, 539)-net in base 3, because

1 times m-reduction [i] would yield (60, 114, 539)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 488862 882833 081300 539924 212058 638571 065105 831800 985435 > 3¹¹⁴ [i]