MinT - Best Known (69, 69+87, s)-Nets in Base 3

Best Known (69, 69+87, s)-Nets in Base 3

(69, 69+87, 48)-Net over F₃ — Constructive and digital

Digital (69, 156, 48)-net over F₃, using

t-expansion [i] based on digital (45, 156, 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+87, 82)-Net over F₃ — Digital

Digital (69, 156, 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+87, 402)-Net in Base 3 — Upper bound on s

There is no (69, 156, 403)-net in base 3, because

1 times m-reduction [i] would yield (69, 155, 403)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 99 048459 789735 697283 815933 291572 395698 973515 104488 922718 426324 133083 339035 > 3¹⁵⁵ [i]