MinT - Best Known (55−31, 55, s)-Nets in Base 128

Best Known (55−31, 55, s)-Nets in Base 128

(55−31, 55, 417)-Net over F₁₂₈ — Constructive and digital

Digital (24, 55, 417)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 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 (9, 40, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(55−31, 55, 515)-Net in Base 128 — Constructive

(24, 55, 515)-net in base 128, using

1 times m-reduction [i] based on (24, 56, 515)-net in base 128, using
- base change [i] based on digital (17, 49, 515)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 16, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
    2. digital (1, 33, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(55−31, 55, 707)-Net over F₁₂₈ — Digital

Digital (24, 55, 707)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(55−31, 55, 1949473)-Net in Base 128 — Upper bound on s

There is no (24, 55, 1949474)-net in base 128, because

1 times m-reduction [i] would yield (24, 54, 1949474)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 615661 080710 231743 018668 865758 684021 828009 496299 831385 725116 154066 441012 057350 676076 229207 021469 761586 005370 363856 > 128⁵⁴ [i]