MinT - Best Known (109−61, 109, s)-Nets in Base 3

Best Known (109−61, 109, s)-Nets in Base 3

(109−61, 109, 48)-Net over F₃ — Constructive and digital

Digital (48, 109, 48)-net over F₃, using

t-expansion [i] based on digital (45, 109, 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]

(109−61, 109, 56)-Net over F₃ — Digital

Digital (48, 109, 56)-net over F₃, using

t-expansion [i] based on digital (40, 109, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(109−61, 109, 285)-Net in Base 3 — Upper bound on s

There is no (48, 109, 286)-net in base 3, because

1 times m-reduction [i] would yield (48, 108, 286)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3401 765140 514883 299449 003797 523522 922866 985650 744085 > 3¹⁰⁸ [i]