MinT - Best Known (33, 33+42, s)-Nets in Base 128

Best Known (33, 33+42, s)-Nets in Base 128

(33, 33+42, 480)-Net over F₁₂₈ — Constructive and digital

Digital (33, 75, 480)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]
2. digital (9, 51, 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]

(33, 33+42, 545)-Net in Base 128 — Constructive

(33, 75, 545)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (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. digital (9, 51, 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]

(33, 33+42, 930)-Net over F₁₂₈ — Digital

Digital (33, 75, 930)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(33, 33+42, 2293172)-Net in Base 128 — Upper bound on s

There is no (33, 75, 2293173)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 109 836799 847092 876874 247067 784869 849980 270308 580367 891312 674513 007567 660317 193799 532178 082456 050783 243281 533586 599719 919611 529956 824984 022334 033134 260115 679392 > 128⁷⁵ [i]