MinT - Best Known (58−33, 58, s)-Nets in Base 128

Best Known (58−33, 58, s)-Nets in Base 128

(58−33, 58, 417)-Net over F₁₂₈ — Constructive and digital

Digital (25, 58, 417)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 16, 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, 42, 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]

(58−33, 58, 515)-Net in Base 128 — Constructive

(25, 58, 515)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 19, 257)-net in base 128, using
  - 5 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
    - base change [i] based on digital (0, 21, 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. (6, 39, 258)-net in base 128, using
  - 1 times m-reduction [i] based on (6, 40, 258)-net in base 128, using
    - base change [i] based on digital (1, 35, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(58−33, 58, 680)-Net over F₁₂₈ — Digital

Digital (25, 58, 680)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(58−33, 58, 1720555)-Net in Base 128 — Upper bound on s

There is no (25, 58, 1720556)-net in base 128, because

1 times m-reduction [i] would yield (25, 57, 1720556)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1 291135 252546 855167 414201 345398 163270 024314 268902 829861 756046 954094 611888 680992 849261 960219 266890 919054 708239 095515 847358 > 128⁵⁷ [i]