MinT - Best Known (24, 50, s)-Nets in Base 128

Best Known (24, 50, s)-Nets in Base 128

(24, 50, 450)-Net over F₁₂₈ — Constructive and digital

Digital (24, 50, 450)-net over F₁₂₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 9, 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 (1, 14, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)
3. digital (1, 27, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(24, 50, 545)-Net in Base 128 — Constructive

(24, 50, 545)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (2, 15, 257)-net in base 128, using
  - 1 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
    - base change [i] based on digital (0, 14, 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 (9, 35, 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]

(24, 50, 1325)-Net over F₁₂₈ — Digital

Digital (24, 50, 1325)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(24, 50, 5678788)-Net in Base 128 — Upper bound on s

There is no (24, 50, 5678789)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 2293 499471 760546 275016 487656 091855 761300 597310 524073 848892 619818 201645 000382 697930 561427 030772 804509 056960 > 128⁵⁰ [i]