MinT - Best Known (31, 70, s)-Nets in Base 128

Best Known (31, 70, s)-Nets in Base 128

(31, 70, 480)-Net over F₁₂₈ — Constructive and digital

Digital (31, 70, 480)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 22, 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, 48, 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]

(31, 70, 545)-Net in Base 128 — Constructive

(31, 70, 545)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 22, 257)-net in base 128, using
  - 2 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. digital (9, 48, 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]

(31, 70, 920)-Net over F₁₂₈ — Digital

Digital (31, 70, 920)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 70, 2804850)-Net in Base 128 — Upper bound on s

There is no (31, 70, 2804851)-net in base 128, because

1 times m-reduction [i] would yield (31, 69, 2804851)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 24 974057 581205 738074 436032 052788 975871 627412 324459 375860 986379 003548 485624 550998 590163 287946 353930 022246 999212 808207 234859 508053 482457 155576 377632 > 128⁶⁹ [i]