MinT - Best Known (26, 26+47, s)-Nets in Base 128

Best Known (26, 26+47, s)-Nets in Base 128

(26, 26+47, 321)-Net over F₁₂₈ — Constructive and digital

Digital (26, 73, 321)-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 (3, 50, 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]

(26, 26+47, 513)-Net over F₁₂₈ — Digital

Digital (26, 73, 513)-net over F₁₂₈, using

t-expansion [i] based on digital (24, 73, 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+47, 293169)-Net in Base 128 — Upper bound on s

There is no (26, 73, 293170)-net in base 128, because

1 times m-reduction [i] would yield (26, 72, 293170)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 52 374660 759480 021157 173854 113259 239739 108570 370529 176831 192763 146511 575562 136894 006535 238228 469622 061763 683283 297460 065165 439325 100299 009244 336468 527728 > 128⁷² [i]