MinT - Best Known (106−65, 106, s)-Nets in Base 3

Best Known (106−65, 106, s)-Nets in Base 3

(106−65, 106, 42)-Net over F₃ — Constructive and digital

Digital (41, 106, 42)-net over F₃, using

t-expansion [i] based on digital (39, 106, 42)-net over F₃, using
- net from sequence [i] based on digital (39, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(106−65, 106, 56)-Net over F₃ — Digital

Digital (41, 106, 56)-net over F₃, using

t-expansion [i] based on digital (40, 106, 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]

(106−65, 106, 205)-Net in Base 3 — Upper bound on s

There is no (41, 106, 206)-net in base 3, because

1 times m-reduction [i] would yield (41, 105, 206)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 137 331598 110016 778072 821360 105309 066264 838855 054145 > 3¹⁰⁵ [i]