MinT - Best Known (155−85, 155, s)-Nets in Base 3

Best Known (155−85, 155, s)-Nets in Base 3

(155−85, 155, 52)-Net over F₃ — Constructive and digital

Digital (70, 155, 52)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 55, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (15, 100, 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]

(155−85, 155, 82)-Net over F₃ — Digital

Digital (70, 155, 82)-net over F₃, using

t-expansion [i] based on digital (69, 155, 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]

(155−85, 155, 423)-Net in Base 3 — Upper bound on s

There is no (70, 155, 424)-net in base 3, because

1 times m-reduction [i] would yield (70, 154, 424)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 30 921915 657516 470383 697823 446875 122944 667702 641689 541253 470809 344840 906225 > 3¹⁵⁴ [i]