MinT - Best Known (67, 67+81, s)-Nets in Base 3

Best Known (67, 67+81, s)-Nets in Base 3

(67, 67+81, 48)-Net over F₃ — Constructive and digital

Digital (67, 148, 48)-net over F₃, using

t-expansion [i] based on digital (45, 148, 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]

(67, 67+81, 72)-Net over F₃ — Digital

Digital (67, 148, 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]

(67, 67+81, 408)-Net in Base 3 — Upper bound on s

There is no (67, 148, 409)-net in base 3, because

1 times m-reduction [i] would yield (67, 147, 409)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14001 165186 300056 041597 981200 739089 229598 563800 416653 043766 076562 318625 > 3¹⁴⁷ [i]