MinT - Best Known (68−43, 68, s)-Nets in Base 128

Best Known (68−43, 68, s)-Nets in Base 128

(68−43, 68, 342)-Net over F₁₂₈ — Constructive and digital

Digital (25, 68, 342)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, 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, 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]

(68−43, 68, 513)-Net over F₁₂₈ — Digital

Digital (25, 68, 513)-net over F₁₂₈, using

t-expansion [i] based on digital (24, 68, 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]

(68−43, 68, 361143)-Net in Base 128 — Upper bound on s

There is no (25, 68, 361144)-net in base 128, because

1 times m-reduction [i] would yield (25, 67, 361144)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1524 331477 791864 726150 738041 925241 533885 690395 724712 326670 414923 620053 304168 148560 065607 222932 530504 822005 018092 514319 729467 686981 125858 461357 > 128⁶⁷ [i]