MinT - Best Known (69, 69+79, s)-Nets in Base 3

Best Known (69, 69+79, s)-Nets in Base 3

(69, 69+79, 56)-Net over F₃ — Constructive and digital

Digital (69, 148, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 54, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 94, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(69, 69+79, 82)-Net over F₃ — Digital

Digital (69, 148, 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+79, 446)-Net in Base 3 — Upper bound on s

There is no (69, 148, 447)-net in base 3, because

1 times m-reduction [i] would yield (69, 147, 447)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14099 582091 625736 713292 213712 065574 350672 136419 201793 843211 225967 884979 > 3¹⁴⁷ [i]