MinT - Best Known (27, 27+35, s)-Nets in Base 128

Best Known (27, 27+35, s)-Nets in Base 128

(27, 27+35, 438)-Net over F₁₂₈ — Constructive and digital

Digital (27, 62, 438)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 18, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (9, 44, 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]

(27, 27+35, 515)-Net in Base 128 — Constructive

(27, 62, 515)-net in base 128, using

2 times m-reduction [i] based on (27, 64, 515)-net in base 128, using
- base change [i] based on digital (19, 56, 515)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 18, 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, 38, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(27, 27+35, 759)-Net over F₁₂₈ — Digital

Digital (27, 62, 759)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(27, 27+35, 2057334)-Net in Base 128 — Upper bound on s

There is no (27, 62, 2057335)-net in base 128, because

1 times m-reduction [i] would yield (27, 61, 2057335)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 346 584532 413267 795248 987596 398139 543571 271764 844847 982043 282672 452015 136682 824370 596907 587536 137519 404295 132503 264886 057970 475126 > 128⁶¹ [i]