MinT - Best Known (27, 27+49, s)-Nets in Base 128

Best Known (27, 27+49, s)-Nets in Base 128

(27, 27+49, 321)-Net over F₁₂₈ — Constructive and digital

Digital (27, 76, 321)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 24, 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, 52, 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]

(27, 27+49, 513)-Net over F₁₂₈ — Digital

Digital (27, 76, 513)-net over F₁₂₈, using

t-expansion [i] based on digital (24, 76, 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, 27+49, 296874)-Net in Base 128 — Upper bound on s

There is no (27, 76, 296875)-net in base 128, because

1 times m-reduction [i] would yield (27, 75, 296875)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 109 840135 609004 774626 367192 919161 777132 258490 871149 913470 428885 303708 445923 852124 527969 528190 870569 071608 722161 034559 035888 804805 495424 388380 255745 053020 240626 > 128⁷⁵ [i]