MinT - Best Known (27, 70, s)-Nets in Base 128

Best Known (27, 70, s)-Nets in Base 128

(27, 70, 384)-Net over F₁₂₈ — Constructive and digital

Digital (27, 70, 384)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]
2. digital (3, 46, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(27, 70, 513)-Net over F₁₂₈ — Digital

Digital (27, 70, 513)-net over F₁₂₈, using

t-expansion [i] based on digital (24, 70, 513)-net over F₁₂₈, using
- net from sequence [i] based on digital (24, 512)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 24 and N(F) ≥ 513, using
    - a curve from the manYPoints database [i]

(27, 70, 573285)-Net in Base 128 — Upper bound on s

There is no (27, 70, 573286)-net in base 128, because

1 times m-reduction [i] would yield (27, 69, 573286)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 24 974339 003996 187339 410133 755291 592215 947046 384446 817568 816950 654485 194259 179323 145164 679498 378013 762367 869442 480984 703379 123580 348428 540971 256182 > 128⁶⁹ [i]