MinT - Best Known (106−61, 106, s)-Nets in Base 3

Best Known (106−61, 106, s)-Nets in Base 3

(106−61, 106, 48)-Net over F₃ — Constructive and digital

Digital (45, 106, 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]

(106−61, 106, 56)-Net over F₃ — Digital

Digital (45, 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]

(106−61, 106, 253)-Net in Base 3 — Upper bound on s

There is no (45, 106, 254)-net in base 3, because

1 times m-reduction [i] would yield (45, 105, 254)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 134 181266 574219 000510 164510 259029 775705 474172 550229 > 3¹⁰⁵ [i]