MinT - Best Known (26, 26+43, s)-Nets in Base 128

Best Known (26, 26+43, s)-Nets in Base 128

(26, 26+43, 345)-Net over F₁₂₈ — Constructive and digital

Digital (26, 69, 345)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 21, 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, 48, 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]

(26, 26+43, 513)-Net over F₁₂₈ — Digital

Digital (26, 69, 513)-net over F₁₂₈, using

t-expansion [i] based on digital (24, 69, 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]

(26, 26+43, 455014)-Net in Base 128 — Upper bound on s

There is no (26, 69, 455015)-net in base 128, because

1 times m-reduction [i] would yield (26, 68, 455015)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 195109 298239 599987 741397 628019 172950 308836 846015 474931 238110 621516 724045 864107 272857 174543 719336 642973 706841 033692 201239 841695 464124 500958 756484 > 128⁶⁸ [i]