MinT - Best Known (66−41, 66, s)-Nets in Base 128

Best Known (66−41, 66, s)-Nets in Base 128

(66−41, 66, 345)-Net over F₁₂₈ — Constructive and digital

Digital (25, 66, 345)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 20, 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, 46, 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]

(66−41, 66, 513)-Net over F₁₂₈ — Digital

Digital (25, 66, 513)-net over F₁₂₈, using

t-expansion [i] based on digital (24, 66, 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]

(66−41, 66, 461238)-Net in Base 128 — Upper bound on s

There is no (25, 66, 461239)-net in base 128, because

1 times m-reduction [i] would yield (25, 65, 461239)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 93036 229976 887895 898566 770296 901468 916932 125118 746077 365780 767853 812897 293688 315787 792416 390829 904300 167910 125117 113240 646374 933727 555609 > 128⁶⁵ [i]