MinT - Best Known (136−67, 136, s)-Nets in Base 3

Best Known (136−67, 136, s)-Nets in Base 3

(136−67, 136, 60)-Net over F₃ — Constructive and digital

Digital (69, 136, 60)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 48, 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 (21, 88, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]

(136−67, 136, 82)-Net over F₃ — Digital

Digital (69, 136, 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]

(136−67, 136, 557)-Net in Base 3 — Upper bound on s

There is no (69, 136, 558)-net in base 3, because

1 times m-reduction [i] would yield (69, 135, 558)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 26787 394663 195561 344610 375517 096889 120230 207423 908598 218742 998045 > 3¹³⁵ [i]