MinT - Best Known (69, 69+91, s)-Nets in Base 3

Best Known (69, 69+91, s)-Nets in Base 3

(69, 69+91, 48)-Net over F₃ — Constructive and digital

Digital (69, 160, 48)-net over F₃, using

t-expansion [i] based on digital (45, 160, 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]

(69, 69+91, 82)-Net over F₃ — Digital

Digital (69, 160, 82)-net over F₃, using

net from sequence [i] based on digital (69, 81)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 69 and N(F) ≥ 82, using
  - Drinfeld modules of rank 1 [i]

(69, 69+91, 384)-Net in Base 3 — Upper bound on s

There is no (69, 160, 385)-net in base 3, because

1 times m-reduction [i] would yield (69, 159, 385)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7388 246006 142768 213789 973446 028304 321395 781987 255161 809065 948710 112915 788371 > 3¹⁵⁹ [i]