MinT - Best Known (68, 68+87, s)-Nets in Base 3

Best Known (68, 68+87, s)-Nets in Base 3

(68, 68+87, 48)-Net over F₃ — Constructive and digital

Digital (68, 155, 48)-net over F₃, using

t-expansion [i] based on 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]

(68, 68+87, 72)-Net over F₃ — Digital

Digital (68, 155, 72)-net over F₃, using

t-expansion [i] based on digital (67, 155, 72)-net over F₃, using
- net from sequence [i] based on digital (67, 71)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(68, 68+87, 390)-Net in Base 3 — Upper bound on s

There is no (68, 155, 391)-net in base 3, because

1 times m-reduction [i] would yield (68, 154, 391)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 30 352948 014029 901799 304615 432487 172882 154288 010738 028474 306830 692981 007435 > 3¹⁵⁴ [i]