MinT - Best Known (75−47, 75, s)-Nets in Base 128

Best Known (75−47, 75, s)-Nets in Base 128

(75−47, 75, 345)-Net over F₁₂₈ — Constructive and digital

Digital (28, 75, 345)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 23, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (5, 52, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(75−47, 75, 577)-Net over F₁₂₈ — Digital

Digital (28, 75, 577)-net over F₁₂₈, using

net from sequence [i] based on digital (28, 576)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 28 and N(F) ≥ 577, using
  - a curve from the manYPoints database [i]

(75−47, 75, 447054)-Net in Base 128 — Upper bound on s

There is no (28, 75, 447055)-net in base 128, because

1 times m-reduction [i] would yield (28, 74, 447055)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 858110 022132 607812 630624 809422 187424 695275 678708 586270 160510 688337 541147 164052 788606 483646 061713 531560 889778 100728 126704 466440 956365 266986 302493 437689 773216 > 128⁷⁴ [i]